{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. 物理问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1 单粒子问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.1.1 经典物理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "考虑一个非谐振子的过程。例如水平方向弹簧连着滑块，但是滑块除了受到弹簧的弹力之外，还受到一个与时间及位置相关的外力作用，如何给出滑块随时间的运动过程？\n",
    "\n",
    "由牛顿第二定律可以给出运动方程\n",
    "$$\n",
    "F_k(x)+F_{\\text {ext }}(x, t)=m \\frac{d^2 x}{d t^2}\n",
    "$$\n",
    "这里弹簧的弹力不一定是线性的。首先我们忽略随时间改变的外力作用。对于弹力，我们可以构造一个模型，使得在x比较小时，可以近似为线性力。猜测该力的作用势形式为\n",
    "$$\n",
    "\\begin{aligned} V(x) & \\simeq \\frac{1}{2} k x^2\\left(1-\\frac{2}{3} \\alpha x\\right), \\\\ \\Rightarrow \\quad F_k(x) & =-\\frac{d V(x)}{d x}=-k x(1-\\alpha x)=m \\frac{d^2 x}{d t^2},\\end{aligned}\n",
    "$$\n",
    "当$\\alpha x \\ll 1$时，我们可以回到简谐运动。\n",
    "\n",
    "从这个势的形式可以看出，如果$x < 1/\\alpha$,则该力为回复力，整个运动是周期的。然而，随着振荡幅度的增大，平衡位置左右的运动将出现不对称。如果$x > 1/ \\alpha$，力将变得排斥。\n",
    "\n",
    "此外，构建其它的模型，例如设弹簧的势函数与平衡位移x的任意偶次幂p成比例\n",
    "$$\n",
    "V(x)=\\frac{1}{p} k x^p, \\quad(p\\ even ).\n",
    "$$\n",
    "对应的力为\n",
    "$$\n",
    "F_k(x)=-\\frac{d V(x)}{d x}=-k x^{p-1}\n",
    "$$\n",
    "\n",
    "首先我们可以大体看一下这两种势"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "def Vr1(x,k,alpha):\n",
    "    return 0.5 * k * x**2 * (1 - 2/3 * alpha * x)\n",
    "\n",
    "def Vr2(x,k,p):\n",
    "    return 1/p * k * x ** p\n",
    "\n",
    "x = np.arange(-1,1,0.01)\n",
    "fig = plt.figure()\n",
    "plt.subplot(2,2,1)\n",
    "plt.plot(x,Vr1(x,1,0.2),label=\"Vr1\")\n",
    "plt.legend()\n",
    "plt.subplot(2,2,2)\n",
    "plt.plot(x,Vr1(x,1,1),label=\"Vr1\")\n",
    "plt.legend()\n",
    "plt.subplot(2,2,3)\n",
    "plt.plot(x,Vr2(x,1,2),label=\"Vr2\")\n",
    "plt.legend()\n",
    "plt.subplot(2,2,4)\n",
    "plt.plot(x,Vr2(x,1,6),label=\"Vr2\")\n",
    "plt.legend()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用ODE函数，通过尝试幂次在p=2-12范围内或非谐强度在$0\\le \\alpha x \\le 2$范围内来研究非谐振荡。不包括任何明确的与时间相关的力。请注意，对于较大的p值，可能需要减小步长h，因为力和加速度在转折点附近会变大。\n",
    "1. 检查在给定的初始条件和p或α值下，无论力的非线性程度如何，解都保持周期性，振幅和周期不变。特别是，检查最大速度出现在x=0处，最小速度出现在最大x处。后者是能量守恒的结果。\n",
    "2. 验证非简谐振动是非同时的，即不同的初始条件（振幅）导致不同的周期\n",
    "3. 解释为什么振荡的形状会因不同的p或$\\alpha$而变化。\n",
    "4. 设计一种算法，通过记录物体过原点的时间来确定振荡的周期T。注意，因为运动可能不对称，必须至少记录三次。\n",
    "5. 画出周期随初始振幅变化的图。\n",
    "6. 验证随着初始能量的接近$k/(6\\alpha^2)$ 或 $p>6$，运动是振荡的，但不是谐振的。\n",
    "7. 验证对于$E= k/(6 \\alpha^2)$的非谐振荡，运动从振荡变为平移。看看离分界线有多近，一次振荡需要无限长的时间。（幂律势没有分界线。）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.integrate import solve_ivp \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def f1(t,y,k,alpha):\n",
    "    return [y[1],-k * y[0]*(1 - alpha * y[0])]\n",
    "\n",
    "def f2(t,y,k,p):\n",
    "    return [y[1],-k * y[0]**(p-1)]\n",
    "\n",
    "y0=[0,1]\n",
    "\n",
    "sol = solve_ivp(f1,(0,10),y0,args=(1,0.1),t_eval=np.arange(0,10,0.01))\n",
    "\n",
    "plt.plot(sol.y[0],label=\"x\")\n",
    "plt.plot(sol.y[1],label=\"v\")\n",
    "plt.legend()\n",
    "plt.show()\n",
    "\n",
    "sol = solve_ivp(f2,(0,50),y0,args=(1,6),t_eval=np.arange(0,50,0.01))\n",
    "\n",
    "plt.plot(sol.y[0],label=\"x\")\n",
    "plt.plot(sol.y[1],label=\"v\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "探索\n",
    "\n",
    "在没有摩擦力和外力的情况下，能量应该是守恒的。在这种情况下，能量的恒定性是对数值计算的一项苛刻的测试。\n",
    "1. 绘制50个周期的势能$PE(t)=V[x(t)]$、动能$KE(t)=mv^2(t)/2$和总能量$E(t)=KE(t)+PE(t)$。讨论$PE(T)$和$KE(T)$之间的相关性，以及它如何取决于势的参数。\n",
    "2. 通过绘图$$-\\log _{10}\\left|\\frac{E(t)-E(t=0)}{E(t=0)}\\right|$$查看解的长期稳定性。因为$E(t)$应该与时间无关，分子是解中的绝对误差，当除以$E(0)$时，就变成了相对误差（比如$10^{−11}$）。如果你不能达到11个或更多的位，那么你需要减少h的值或调试。\n",
    "3. 因为被大p谐振子势束缚的粒子在大多数时候基本上是“自由的”，你应该观察到它的动能随时间的平均值超过了它的平均势能。这实际上是幂律势的维里定理背后的物理学：\n",
    "$$\n",
    "\\langle\\mathrm{KE}\\rangle=\\frac{p}{2}\\langle\\mathrm{PE}\\rangle.\n",
    "$$验证你的解决方案是否满足维里定理。\n",
    "\n",
    "后续大家可以继续思考有外力及摩擦力的情形。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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suHkQqe+FTvnF+boJFKPPbCeUTJ9CDxToP04kep1Opmz4pnMvEozkhUzZKMqFGzPLdNLGBYQego7iKK0qjcQ5pAA+FzpW/G4Aqq4X8aKspJSV0LwwHDrH39HNrVXenuJadqOKX4prPXMPu1Ay9Qqdnu5cjDLgoQlF3XDDFhV6TKiRO0ppKTqKfDCkDmOOUyptJzWdOQki48phFyrjjxu22O2EcBjzAEVcWTdcByJ7TyUJoogPBFILMUX6ZISeR6ekKpC8p1Q7LYd+cUQtWvWgONlxCgVQH3ban7SdqCPoTsic3+VGK1SiZRj9yQ87dhx6NsdcRcxUWtXEIg7lorIYWUW+Ej7Srzp6OeAGmOwpTn+6Qai6JNtT9PrsoTj06p5q49ADS82ByFAU6cbkJnZMMk65qdfBUrjZfHPm+GOiUMWhc+a4yqGzHX9dfjsTOipEYhsjpDPg8wb46y8dk6IfE3KOyCjOfC8Mv5SylntRB0LQC+fl+Qt9HkgKJU4KXQhxmxDioBDikBDiA5bvvUoIEQshfjRcF/1ksvgUKuE5t7jlVFUSBC9ssYKQ2NEyfJ44tzoCOIw5czwJ6UzHxInAAPhAQMVIcxF6qQAVO2Wftxeq9AQ5/FGFu3Y6rMqExUADrhO8y6SAajH6lzpCF0JEAD4C4E0AbgbwTiHEzYbvfQjA50J30kfGVeXHMDMn8ePMxKI8FZxJuQSgkYAwkRxK2XCVVjAOnYsek3DKL5TDOI9n50ZHhULoTOqmGD9ebNdXRnGSBRrQLUT1tyPm4VvdU5cDh34rgENSyqeklEMAdwB4q+Z7/wzAJwGcCtg/b5koP36stTq9qQ9b1RmJOrzEoqrjhcuh81P/FULiFvlKCgidd9j1mQf4KC7PMUf5dTOqTkrQS82qOjccDj0bQ7fD4/QnFiK/LIKiHzntlJP1eOsmrSVEp/xy/wLzsAslLgr9GgCHC6+PZO/lIoS4BsCPALjd1pAQ4j1CiANCiAOnT5/27auT5Ag9BJrodLICQDxFnCaaMBZNQBqp2A4vsajDqkSpDruUQ+cfdl1m1E01yoVLB3BRaJGO4iaSqTnmJNkBhYgQJocedTr5a4oUQ4G5VAmXHqtbvZe+Qhea96q9/jCAX5FSxraGpJQflVLul1Lu3717t2MX/aTmwCGH+CVsDl39HtdJVuXQqVbdOJboCKDH5ELHWWJH1BFIiChUPRel/Lhz3GO2Myk4xou1HhX8C8X+efcnSRPbOA7jotLqdQRZgdYLUNHpsaiE0Onrj1tOQ/3tSRgl07/AjBwLJV2H7xwBcG3h9V4Axyrf2Q/gDiEEAOwC8GYhxFhK+WchOukjVccfZ0PltEKSVm3MxufRRoaQ8iQInunc7/I88sX6IOo1RYolYlU7/Y7f3EwOOxVJZMUCRlGKWPHNXJ44BELvd4tznACISP3pZVYQt6hb7pTnWppd/iGlgECxXe924kL4I1MRcxF6Ldx1g52iLgr9HgA3CSFuAHAUwDsA/HjxC1LKG9TPQojfA/DnG6HMgcki6TNphdyBky2+RAKRn87K/7aK2V4f8agSVTucw6GrULhi/7z7oxI7Ckqr7xkBm9MBTCtoQrkohL7RVRsl5jMFCtCUnyrtGsr3wnXK19YNFwgw28n9WzF9L4yKe5MVgRYmOiqUNCp0KeVYCPE+pNErEYCPSykfFUK8N/vcypt/q0U52hR65Jy8ypkEKHPRD2mV+d0OxvGY1JdQHnlVKGyibDgmb4eFtOKCIuagxyLS73boiU6hknCK3HexXd82gKIi5jpF+QdDEdzQY7Yr64ZxN24KbiRbEfei1E8WKrfjklfoACClvAvAXZX3tIpcSvkz/G7RRSHroiKmtsPlQssbk4esVTscj7xKluJGGaT1QQqUC2FTFdEjJ8qljPQZ5RWqJWIZ/VFZjABt3eQHeMSM5CgcmtyDoayIeeuP6zAu5ohweHhgQtVxAcUMs5ZQKJm6TNFRtvi6DEUM1FEJ5YGPCvwuCyHFk43J9cgXDylOf7qldvwXcZlDDxCtwIzkCBUBFAKh55x1hxfJMc4Puw7rYCjeDaBek/qTTGLr0/4xKL8ojDOdHeWSzWmfmSMSSqZOocdJUkHovI2p2iFFchRRKCexqMahc0zVsjOT2p8uE+kH59A73PIKSf6c1GuKFDODAVpoXlzgd0M4M/k1YZLyAc52ZoawntVdtIE4dFbtnolfoK3lElhGOULi394edXgRIbUYaebG5HLoVRqJx2FOaAUK0ioja/6NRV0m0lfoscdVWglfaZXT0uk3H00sO95lEMWsVfWa1J+ksqeoyjhEPfQCSAq5pzgXvoSQqVPo45ozk85b9yIeF1pC6AylFVfQLMe5FYLDnIQtcpRWGSHRTd5iOzzl14v4PPE4QBx6qKim2sHAmOPS3DD3VJdhISaJhJSTuaEXLgtTy6VK8W60U3T6FHrBYQKEKc6lXvvKqBQjzTPrgMxJxqQVyhwmvcgSd4OX0tJDZPwF4FTVPakA72KUbkcgEgEsu5yq40e5cC4RCban4sqeIszxqKiIL4FMUUXxdjoCHbHxcejTp9DjSvJCIMcfiQutcJhcnrjHrjsRKC095ju3ylEG/Ov5umxONclvCAL4BahY0VG1GOkwc8NPsgtDR4WKHIuy8s2UMrxVGpMV0pmNh3P4hpKpU+ihPPLKucXjQsNw6HFl8bGr3QXYmFzlN4kDZjr+inQUp/peJUafp7R4tFYRhXLKK9S5eDoKLSF0VrLexJlOyRmoOjNVu74SqnCZWjdpW7zKrCFk6hR6irT4fF/Vsx8ETXDDFgNQN1GHl8U4aSelf9LXG8ihF+koFqeqLl/g3+5TjkOnh3SqyyAAWk2iKq3AvaEq3J7izw3XFxSXLEReBJqyVLklk0PI1Cn0CQqlb8xi2VtWlEuRQ4/oMdJxkkAIoMNE6HltdnYccAKuI6iMkAIVoOKGLUYpD5q+Zka55IcmoY1S2CL9WY0Kc8zlm9Obfbgx21UOnbmnAjielYXI5dABRbm0CD2ojCoolOuUCpHxp+gJcghlInMlzIp6yDj0TkdACHoccI1TDcKh8xC6Ohg4qf/djoAQPE4/RKay+p1exIskikuUC2/9qfFEghcaykXWZUVMj5apVkKlVmUtzg1njkPJ1Cn0OEnSUqoMj/wkcoKHQqt1RrhhgmlbfA4dSB2s/MQifkhnESGRnFsFOopzx6SyOgCQnWRSylqsNSv1n3lojuIyMOFw6N3S+uPe5kRXxBPrpRjPzvBvdZjXQ8YFpyjDKR9Kpk6hVwtQ0VDAxNHGSRgoZkNybrAZlxYNfWMq6wVQSoubWBSOQweoB0OZjuJW1wRArntS5awBHhBIqRI+ClXAJJjjj4r0CzdUpe3yrBdO4EOVDuVFAGV7inklYwiZOoWu+D5OuOFY40nnZEMWo24oPHqcJLkDks335SFWtI2p8y+wowxYESETOopTgGpUODSpc1wshpX7cEjIukyVpG3TAYVyYNN9OJPDLooE/cYi5d8KQLmw/VvVgAWm7wXgWb2hZOoUeh5lECKxg1nXuup4obYzKnrSOQkiFfOQUyWxOCaKmRknYZxbYemoop+CE50SCqEzHYiFZ9Vj+im4kRxlOoqRv1BxpgP0ddPJLLuQdGgbthhYRllikcrcSpjhXrzQKA3fR2knAHpUf7vskeeNicOhF5VNh3P41jhMXtlbgM6hj4vPm0MHaCw7ruOPWy+Hu/50e4pCY1ZvAQNoWb0lqoRJR+UBC21iUXgpnpjUrMpy2Vu60qpWFASIFFAShsNUVerydpgcJodDr0YZFN/z7c/kkOIhdBVRQp3jcVz3vZDmJlaH3QTNcnw46napMBw6bU/pqBKKZVeitRiHZlzbUzxHb95Oi9DDyihJctOZiyZ6EY+6KSEklnNrorS4ceg9JtLS+Rdoh12RQ6eXrB0nRUcv/U7RMq1F41RLdWWYhxTAV35xIicOYwZVV+LQA+ypLseyK8bWM6xe5aAFeHtqVAEULUIPLHESjicuZ1VyUv+Zzq2kEuVCRaEFpEV1IOr8CzSHcaa0ive2EvRNkY7iXKhcchhTEXogDl3XDvXQDJG/ULPsWAh9oogpTtpxwWHM5dAjpkWm2uHqm5AydQpdXZcFqDAiJk8XgMMsoQnihQch+L5RBWmxanV3mPHECrEx6YlRxXphFWMroVDeLUycTOWiZce5jETVYAHSNUguPZHIimXHUMTFQyrA1YVACA6dV5ZjctjRo2VCyfQp9BBIq3JjDMDj0LnFwuocOsN0LnHoHJOX59wKxaEXERLr3syKk4xCcYy0HDqdjirXZ6dw6DIIHTAO4DDW0VGsHBFmBFopISjAbU6qHWpoaCiZPoVeQ1pMvo+pbFQ7vDTlCQrtdATrIuSIuYj1NBKTQ2fSEyVlw8kULSQWBePQSfet1oEA1UIMHx3FjNHviLwmDMlC1ACKEOsmxNxw2gkl06fQE5mbqXyPfJiaMOUkCKp5yOf7yu1QwxaVsmEipBKHzqAnCnRUjzk35YOByX0HSJ4p0VrEejnF2HpOYtFkboh7qpCyr/rDBxScsNmCM5Nt9U72VJv6H1iq5iEnLbi4oXixtzwH4jgQhz6uOoKIZjyArF4O/WJcrQORrLQmyoZTXqHoQORz6AGyaLn1+OMJ993lzE2FquNE7pRCQwOtGzqHPtETCWPdRMx1E1KmT6EXzUMiCgiV8acUZkdMFnKImFlqZbg4KSZB8Oam7Begx1qXYvTJymZCBwDE6InCYbfRHLrWsqPmL0TluaE+cy6tVVw3aX9otNYEbPEOzaJll7dDuvmIX04jpEynQud65ANxoSn9ozhDTkRIGYVSOHQpZT3pilGlrhd1WDH6o8IG51yUUT3s1Hu+UozRp/LEZQ6dY9mFiyTqFSw7an9GcZme4OUdTNqhImuA718oU2w8K7zLtDpCylQpdKW0uGVQx0WeOBCyCYlCubwswKejos6kvAJVaUWdSf3x9D2qsikrLVISTiX8jIuss67weWIW5ZLU1h+tOFwZJHFLRgAM56o2i5YXaMB17k/2Zhu2GFSqSou7aLh0wKjCfadt07jQiMmhT9CjqkzId0oBdEdQ1ZIqtu0j1bBF9Z6vjJKkxO9yr47jXJShy4YkA4HKYedr3RWLaqV94nHo3HVTjAAKGR0F0OLiq0lX1FpCoWS6FHpcVlpUj3zRdOYpmwlC6jGRfjnKhVPQnxdPXFQ2qh0qh16MDgD4MfqcQ7NeNY/OoYeKlumIydyQDs04KaHHYtuuor4+WTe0Il96sMVwyhfDZokgqehPSvtIS7TrRTwAGVKmS6EXuEf1Pw2t1cPGqMqmV+C+6e2U66FTPPLjirKh3n9YQ1oMpF9TxNTolILVQWlHSpkl4YRxGJeQPhGF5r4XVmVCWaMVfNdfsa5M+j8vcqf4zCnXBRYtzZCO3mLb1HZSq6NF6MFkgtC5HvkJh87hQnUcOlX5dSvt+Hrkxxplw4kn5h6a1RhpgK60qs4tXwQZV9EjMYW7yKGr/6lznFt2zMOuTmv5zfE4rq4bGhCYHHYF6oZiBWkcxiHi0NP3iECAuRdCipNCF0LcJoQ4KIQ4JIT4gObztwohHhJCPCCEOCCE+J7wXW2WGq0g+Bw6hwsNyaEX604A/ou45pRims7cRJOqow2gheYVi2pRL1Su+wW4F1zwqjZWyxAA1Lmph+b5tjOurBuyM72C9MmJbaV7Ujc2yiWnoxTFG9GsjpDSqNCFEBGAjwB4E4CbAbxTCHFz5WtfAPByKeUtAH4OwMcC99NJinUe0v+ZcegRD2mViyOF49Ap7RRjpNX/vBvpmXXV43r8OPXw5ZrO2gigDeTQS6FwjOJc5dKutHbq1gstD6Jm2RHL+RbDXbkc+uR5Z+8x6ajLBaHfCuCQlPIpKeUQwB0A3lr8gpRyWU6ubN8EYENGpePpOBx66UFRIzkqG4rs+IsqSotIKxRruXArAap2yAgpQNhYtdQD4O9AnKC+CRDg8Lu530RQLbukfoATHb01qo7NodP3QtoPXrSMssiE4F9wUbd6aXRUlaqTxMS/EOKi0K8BcLjw+kj2XkmEED8ihHgCwP9EitJrIoR4T0bJHDh9+jSlv1bR8sSkmOTJTS8AE2kF4dCTGpoNQSvQIieyOWaHsfFpJPU73Fj/UcXRS6cVwnDopTExHH+jSrhr2g5TaQWyeqkOxOrtSQB9T3EBxWRPVWitDUTpLgpdaN6r9VhK+Wkp5YsA/DCAX9M1JKX8qJRyv5Ry/+7du7066iKTDLvJIqbcKZpvTOYG120oeogfL+OvHm5I5XcnRbVUf6h0VIjDroRmian/Ne6bmkWrQfrcw453oXI5tp7STjV/IRiHzpibXjW2PpAPh+qXqoaGbiTt4qLQjwC4tvB6L4Bjpi9LKf8OwPOFELuYffOWenox81YetuOvvqHISThsNFHeUD0ih5lbQcU5ZiIt1mFXmhvahtL5F1gVJJmUX3FMaTYtfW64oaG1+HFmtcWi1UuNra9lvxKLzHEtu1ooMKM/ocRFod8D4CYhxA1CiD6AdwC4s/gFIcSNQqSFPYQQrwDQB3A2dGebJBSaqCM2ek2YKMDpXd6YvCiXXhFZMygXdkXLAIdU3k7FevHdUDUnODOLtpxxSrM61FhUOxROP81+VTH61HUT7pACJpYdteZ8Mbejkx92YRA6+7BjcPqhpNv0BSnlWAjxPgCfAxAB+LiU8lEhxHuzz28H8L8B+CkhxAjAGoC3yw3wDASL5Kgg/XQR+/cnxKIBsmgFJj1R5Xe72e0+UkpkZ7FXOyFKCHBpJPU71Wxcf4Red4Lz6pVwLbvJukn7Q7xwQ2MFkX0vzD1VjB9X/eFy6ECK+Mnrr+JMJ1MuFQ59I2uiNyp0AJBS3gXgrsp7txd+/hCAD4Xtmr/owg1pHvkkvy1dtUetyVE36/zaSRIJKaFRflTnVjWZZ4KU3dqpb0xOtEKxT9R0++LzprRTT3qZRCv4HHYjzRxzOXTVDolWKKBZOq0QxiKrHna9AP4FXn+SOtjyXDc1AHmZcOiXjdSQFgs9TjYyvSZHUnPg+J7euuiU4vs+fSm1Q4yeqCF9aoJSET2yw88mZnyxj65SjR+nKj8dPUHjm5PSIRsxQkzrdW5oz7uUdxCEQ6ci6/LcUA+7Mh3Kc4pW5/hS59AvG6kmvbCcUhWEtFEhdaZFw09vpyn0KpqlllcIwaFXKwFSrZeaf4F52JWdqzxlk7ZDU35pBFCZQ/d/3tVDinbzUTU6ilNkrgS2GBVV2WURKuuGc4lNKJkyha5Bj8yHnbZH3VB8Dr3KPVLRRNV0znlDX8RWuIVJ9Scch04NNyxvKH8raFK7p9ge5bDrFKg6quM5TiaVAFV/qIlFvQodxU6eCXjYUUMxy2CL7qeoPm9uOY3JJTYtQg8iutAoasxsVDHruAidWhMmrjloaUirlqlHjNlWUQaKW+Zm/AEg33xUpaOoEUCTdVNpx5tTrVh2DPRYQujUQzOuKy2qRcYtr1Bdx5wEpSrlQvWTVQ+7EOU0KO2ElKlS6FXzkGfWlR0vdL6P184kyYmLJurZkJR2qnQAuT52QWlRw89MNJIvh1m17KhFvoq1e9L2wvHE1Ju3akrL95Ay+nB4c0yN3BnVrGf/PZUkEonUWKvkKJfK3FzqUS6XixSL3wMMDj2uho3Rb7CptsPlvql8XzXpqkdF+rGGDmBy6JN2qDRS2eogF+diZiDqxsR19Kp++bZjUlpUDp2dqVy4clC1Q3NmlkFSU5Gv0WiEI0eOYH19PX9PSonfecvV2Da3jscffxyjOMHvvOVqXJGcxeOPLzj3pTuO8TtvuRpb1k7i8cfPYHecvo7PHcHji8bcS2eZnZ3F3r170ev13PvE/quXkOjiQqlx31EVBZBC6qqms//GDBeBYXKu+iKtMh1FTcKpOp6jjiA72oq1ugF4J+FUQzE59ESI6KhiyQjVH6rVkUenEC/KqPte6PRYcW7IYYvVPdUwx0eOHMGWLVuwb9++/DBJEonxsQu4atss9myZxWAUAyeXcO0V89gx33fuy9L6CJ0zK3j+7s3YNNPF0voIUeE1R6SUOHv2LI4cOYIbbrjB+feminKpclrKI09Rfr0i0mKEjdUQOjcCI1T4GcO5VVTEPYafopo8410lsWbGZ0qLqPyqXLx/XHJdEXOrJKp+URAxwFfE9dwO4iUiukADYhmM0t7sdKwc+vr6Onbu3FnKJ5BZKSoB5d/K3mcyJeovhCBchBDYuXNnybJwkalS6DVOi1GZMAwXKmscOpUOKF5MAfBjpKnJPMV7KlW/aBUt644/X/RYvCoQYIQbVhKCOBx6FaGTD7uoPMfeVoeBqiPTUdVDk3BjVlRB6FSqLvL0b1WTw1TXJ2+nP0hPVVxtR4Q6GVSvPJLalEyVQq/WMadXJqyGRvEjOVQ7ZBQqymPicuihDjtyvRIN3+yrtKp1zMl+gYvkMO5FYVL/ewQfjq5WN8AP6eRQdb0KuEkk7W7cXgUI+K7h6l/M9abno1Jfr6rdjXOJTplCr9Yx55iZNQ49AN9HSYLIaaRK8gLddK7MDeGA6dU4dCqtULFeiMqGrYg1Nd7T9plzQ6RcqtQNpTKh7vautI8bE4de3VP0RKcAgEIha5T/993hqlyVQtImgB5FEW655Ra89KUvxdve9jasrq6W3lf/fv3Xf92zB3WZKqfoJMSPF0ZU5+noGX9FVEIpJBSaQ69Wf6TQE1EFPVLT27nZkPVQuPR/fwditcY7lY7yc9iZROfDWR/xFDE11roemkf3L1TpqLQ/CfoeuHIUl5E+pfREzqE3KGJfMR0Mc3NzeOCBBwAA73rXu3D77bfj/e9/f+n9UDJVCD13/BUu6U3f93dEhohy0bXjjZAqJi8V2RgjOdims90pZW6n7vjj3qbT6Qh0BJ0n7lXnhvCsepWQOmoBKu5hZ7LIvJOlTNFRlD1VAUlpO3zr2XeOq1SJCMyh2wrNvu51r8OhQ4e8/o6PTBdCN4T4Uaqo1Th00s1HdScZBfWpPqg2AILpXItWoNMTQbIYKw5jCpqdxOgXlah/tEwtNI9RXqGqiMmx1lWk79lOPeIrs168/RRlGpNar2Sc1PMXACLSr1ov49jpd//NZx/FY8cWkUiJtWGMmV6U92NlMEa/2ykdyE0yTiSet30W/+FtL3f7/niMu+++G7fddhsAYG1tDbfcckv++Qc/+EG8/e1vd/77OpkqhV53/NEzwKoo1HfhVRM70v5wlFaVVuApPyoXr4vcodWcr4ctUqwFAJoyDdTDruL4Y8Zak53p1fBHUthi+QAXQhAtxDKNSa0aWsuiZVjPvarvhTDHgP5uTS9x5NCLivt1r3sdfv7nfx4ALgrlMlUKvVociZMKPtsrh1hxEzvS/vibzrXUa+IhVS2qRS0kpEWPnn0xHXbckDr1sz9Cr9JRdCBQOuwi2k1Dulhr7/UX6+eGW1SLevNRNclO0VukjNOoCgTc2vjVH3oJAGBtOMY3Ty3j+p2bsG0uzcJ8+MgCdm+ZxVXbZp37cnZ5gKMLa/nrCXVTlouhuE0yVRz6SFM7BaCZztx44mrSC0BDj6bUa8oB0+2Ui2oBNMdfFVn7llNV9FXV8cyNHwdo2bgmOsp3jqtXx/UYF3HzD7vyukl/9o8kqlp21LDFutXLCFhgWkFVDj19Iegc+qSJ7H2md5Uh04XQ43p6McCnFXqRP0Kqlr0FeBx6tXAUDT2WlQRAm5vZXpkOUO/3O8L0a+U2KjVY0p8ZdW5qBwPvNifOHM/06nQA5Zq/6rqhXoxSKtNAODSrXDxnT4UIWxxX6KjUCvKMQ684M4FMKXvq4fxgUJRL5f0mqXLot912Gzt0caoUen0jENOUNbQCtVxoFWl5h58Z0CM3ZZ9arEmXqQf4Kb9qDRYA2R2ntEOzFipIeN5C6CI5/OdmXuf4q1AoNsmvHKyEu3Jj6wHaHMeJLF3HSKbqahnGnCir8AhdCEIcel5CoPJ+paHl5WXt78exmzPXR6aKctGFjan3vdrRJASRU6+rzlUyh1lO/aek7FcVn3rfR+IkqZm8gF9ddRMdRakPApSVFil5q+bMJPopqkCAwDdXY+sBWvJWtZYLQFN+oaJTdKn/6n2vdjSRRN4hxRqILiC849Bl5WQQ1JTTgDJdCl2T2AHwoxVICElDuVBSuONKO2kJUlrYYjXuW73v1Y5hjn242Wo0UtofymGXOXoLq7hHqHuiy1pN++kPBKqWnW87pnVD9S/U/BRcIEAO6aznLxT76SLqysFquGsYDt0/Dr3wq6X/N5BCnzKFXnG8sApQaYpq+Tg7dFEGFJ44V37MuifjavwudW6qzi0CF1pNekn7Q0Gh9UgiSmjeqBZCSY3A0PspfNqZPG9e2KxujinWi67GO0CIZzf6cPwtu9KecrBeqvvWxKH7I3QJAVEPW/Rrxtq+r0yXQjegCYqiKCIbinlY5b7Vz/RbecoHFWljarlvfy5Ui9AJHDo3G1cbmkdILArlMI6TevlcwC9mW6e0KFm0VWdm+jPVetHRUf7F4XSHJntPNVgvs7OzOHv2bEk5mjh0X5GG3wuB0FU99NlZ9zBKYBqdohWvvnrft51q+VwgVSC9yLGNfEMxOfRYz4VS6pV0mYeU+r5ujr1oBY0iplx4oI/1pxRAqypiOofe06wbr8OuUowt7U+4CCCKItY5wUmHps4p79GOPu/AXnpi7969OHLkCE6fPp2/tzaMcXZlCCzM5Gvn5OI6elEHyyfdL7hYWBthdTBGtDiXv3fq/BpWZ7s4P+d+y5BJ1I1FPjJdCr0atsi42b50iUMJ6btpdN3i67FohbKZGSK2Xr3PaYeG0AM5jHWHHSEJLK5ETpCBgKYsQtqOD4euiU7JUKhP+GO1VjxADQ1NKtYqMbGomhBEoVy0eQd2hN7r9Wo3/tz54DH84p334/Pvfz1u3LMFAPCL//FvcdOVm/Fb73q5c39+9TOP4M8eOIMHf/X78/d+5P++Gz/96n344Jtf7NxOSJkuysVAK1CUqNY89EETWi6UntjBr0xYPaToYYvlOx392zE7/nghnapNfg0Wakidnm+mrJvq1YUA4DOsakJa2g7NCqpSY2k//R3GPV07Pv6FpG690JL+dNYzLda/Go5KuXkrpEyZQg9EK8TVFO7scgAvpKXhMAm1XCYXX1cQGyUCQ4dCKdQNF6FrlBaNQ9dlQ/o7EGvOdE7yVod32JmUFuBHa420c0y4+ShQsl7Vv0CxnvX+JP/EIt3661EyjCtAAFCHpn9to1AyXQo9FK1QNTNZtEJFaRHRBLfIly7cq9hP5/5UY/QJVlC1xjtAH5P63XI7/odU1QIqtu8q1dR/ShlendKiZK5O5rjqX/BPLNKV06BUMNXTWj6HlN7R61t6wrQ3aeU0yiqUEoEWUqZLoQegFXSFoyiLWFtnhFRRUOcIIqDZCrJW1fdoEUC8RJNqjfe0HUYNFm7yjAEI0Kp08tZN9YaqtB2CA9HgTKeVkuYfdtVIIgrS1zp6ORFozGsmq4yAapNyP0AomS6FHoBWUOZb6aYhDk8ciN8tOsMovGFVaan+ULh4Hb/rNTd5hicTIRkiiShcqDbWmnD4RoGQtZ7mIDhXA4TN6hOLKLH+PKSvp5EolF8YOrTqa1P98aWAQsp0KfQAtILOCUk5GHToMSIumipP1yPU5KgqLdU330y9UVyvdgfQlFY1FXzDEHrFdO50VDYuJTpKg6y9aAW9M131070veg49FFVHOjQrnLV6372NOthilZ6oleH1BxTVCzEo6zikOCl0IcRtQoiDQohDQogPaD5/lxDioezfl4UQ7rE/ASVELQ1dWjopsUhLuVCQdTm2GaA7EKt8ny/S12YfUrhQLYdJcGbG5fr3qk1/J1lSQ1q+978qqq5EK5CoOnVDULEdivIzONOZ9e8nF2UQgAnTetZGAClAQYhnLxcu40dHqb5RrqsMJY0KXQgRAfgIgDcBuBnAO4UQN1e+9jSAN0gpXwbg1wB8NHRHXaRWVItgOuudSf4cZqyNVujk8cSuokPotLondfPQF+nrCkdROHRd2FiP5Mwso0fVH256O+B/2Gktso20EANx6CalRTk0qwoUoB5SRauXQvmZkvV41kvajr/1HFJcEPqtAA5JKZ+SUg4B3AHgrcUvSCm/LKU8n738KgC/9KZAYkrZp/Br1Wp3afsE07lTR2y+qFintCgVJHVKi5Kpp3P8UaygqtJKfKMVNMi6G1GQft168VV+E0XMDVusW0EU5adLLKJy6NVY617HL59CZ71MQJIP2NI50xkRaBVnL7e6JpBx8ZcyQgdwDYDDhddHsvdM8vMA7tZ9IIR4jxDigBDiQDEVN5RUNyYteaGutCgmr6k4km872kVDjXKp8X1+SF/nhKRwoaa0dN92dFYHtchX7bDztBi0Nd5zZeOvtLSRJZ7UTbfmTPdXWqOKozdtx88XpFWglEPK4ExPP/OzwqtUHalEcaBAg5DiotCF5j1tj4UQ/wipQv8V3edSyo9KKfdLKffv3r3bvZeOUr0RvEehSgy36fi2o4+ZpVgMmkVDrZqn5fv8rQ49Qqc4jOto1rfIV816ISCktHCUznTmOiH9HYi66poUP4WWqiNQJdX692nffKk6RT/yAhZ02deUPaV/3rSAhapTlHIFYkhxUehHAFxbeL0XwLHql4QQLwPwMQBvlVKeDdM9P0mLI9WjFfwcdnWkxQkb48YT6ygXesafRvmRQjG5G9N82Hll42oREq0mTD2F249W0NbuoRxSGmc6hTocaSIwKKUntNmQAf0LXmPShQKTLER9/Dj30g6AFn4bUlwU+j0AbhJC3CCE6AN4B4A7i18QQlwH4FMAflJK+Y3w3XQTHU/se/LqMvVIiUU2B6JnREg9AsM/4y8EdWOq8V78zK0v4aIVagiJ7BTl0Qq2CCDSBRfa1H9Py67KfZMzjDVIn7luFPCiVOnU3aDke8DoDinKtXo1/8IGhy02VluUUo6FEO8D8DmkpQY/LqV8VAjx3uzz2wH8awA7AfxWxtmNpZT7L1639aJDJb7OjpHGKUqKmc3Dz3g1QuJKIo/qG6lKog6VEOaGm3SliySiRivU5oZQbXGkCQ31LSGgXzc0ig0orxsS0k/qFhmtkJXOsvPbUzrLjhLrrystTC3fHILGrCZdpf3pYJSEvyvUVZzK50op7wJwV+W92ws/vxvAu8N2zV9MTjISCmDGzOqTF2i8IdeZqf6m1iPPvAuUxKEbwsbSfvrRWjr0SMqi1UUAkQ4pnvWiWzfUsEVdJUBKfZpQe4ob668LWKAidF3kmG/Y4ijWBBoQrOeQMl2Zohrl53vy6ugASmLRSIO0SBceBHBmAlnUg45TpYwpQHSKsR1fWsHAofvF+us2pl9Njsm60eQvEHjiHnP9mWLr/S8FN1UU5DkzKf3RVtekcugaZB0u0ODSdopeNjJK6qjEt7bHWEMrTDYmIfyMWSPEzPfxudCepyNoUhKY51/Q1ngnh3TWkVbxb7hItZyB6hvJYawZk5/1orOCKCUEygECqj/c1H/VH689pZkbSn8mBwM/cqxeToMatsi3nkPK1Cj09BJnaCbYzwTSXkRLCls0x9768oY10zlk2CKpzkhxQxFQqKESJeAf/lijXIhcvN6y49EKlLonk+goZkVLQ4AAtz4N4L+ndBFfqj9ee0pTy4Ua0qmLAOKW2Z6001IubNGVHQX8H5SuzgjV8VetkkilJ7SOF8+iWnrnli9C0lgvBBRqt164lAuBHjPRE5TQPE1IXYhKgMW/4dYfXe2eMKUnvOdGc4ADKp6dFy1DLd9sKqfhMzdaOopgPYeUqVHoOl42fc3PhqRlnOqiU/w3eKxBE2SExA1btFkvBIQeiXo7IRKLAIIDsWrZedJRtrR0v5uuzLSC7xzrYuvTvvquP+66MVAunmh2AtqYCWkWIOBN1TF9dqFlehS6RhED9CJLWqeoT5SLhpclcei60Dxq/Lgmq40StqjN1PPk0DuinHpNCw3lI3RdnRGAsG4saelezlVtrLV/3RPdYUctw1tN/fcNBTZSLsSABa5TvnrZRrFvFCu8KJR6/CFlahS6LgID4IQt1hE6N3mB2g63XoSuzkjeDiEUU4fQfTJXTYk8gK/S0kU1+WXj6njZtB3fWi4aRy+JYksgKocd1QrScd+AvxWk8+H4ZvQCGsrFM+PZZj37WKy6+jQ0hF7POPW1nkPL1Ch0XfJC+toXBdT5XbXZfT37OiWRfuaJJriJHZrICUo7OqeoECJztvlZHbqIB4DCE+sPTVdlbIqRDsGhU675GyeyTv8Qufi648/PCtJdxwj4O1d1CUFpf/zWjS1s1mdvmsIW0776PfPqs2rDFgOJeWP6ogCb48UPldQ3gv/GHMVJKclE9YfGy2qSKUgVBfnV93Tx4wA/Dt0XhRrpKM+NqavxnrfjqfxMlp2/9cLjiU2+F/+wRZv17O/MLAUaBOLQfes1qUAD/W1irUJniy4tHfDP3NIhfarJq1N8vu3EiY6L97vp3OSU8i3WZDo0fQtZ6QuOhQo/87OmdMXYAH+eWFfjPW3H3/GsM+PVZz790fmTAPeDQVefRvWHsqdqNAchQSlEVJMp6cqnHdMh5WuRhZapUegmx0vkjQLqSkvRCr6JRbrNDfghrZHOKRX5RU8YU68j4WW9mA5NShKOLrED8KcVTHPsjdCryo9YFoFbTlWntEgF0AyFowB3pTWyAAFuUbf0ta9T3gySvPMXNFFN6jMX0V1okrbThi0GEV39cSDdYLTypTxOVRfbTOfQec5V22EXJlrBsz62ITMT8E/CMdEKro5wU/4CtUSstgY5MzqFdqGyPsMTcF83sc3qCLFuPJ3yY23Z21Acuh/YmoCbOh0lpf8F46FkahS6EWmRF58mpddn0RjqjAD+UQY6Dr3Y18Y2DCGdvjedm9vxV37VMVFrh5ucq76Hnc50ppSM0NJanmMyHXa+MdumsFkuQvd3phvWjWdim41G4qb++4ItXcRXuZ2NoV2mR6FrolMAAgoItTG1mXo0LrSvcdilfeUpLTLSYvLEpssX0r/BQ7O+dXdMh1SPWHa51+XV9tAddvQqnbyDQT3Tfre+/mgXr2soF24FSYL1MowTzZj8AEXuM2HuzdAyPQpdEz8O0JMXdI4p36Japkw9X6RVW3yRr9LS833k1OuqovCc41Gc1A4pynWBphh9n3ZM+QuRJ3ocmtrxphXqoXC08EfdYeeJ0A0+E2q1RXbVRmvkjueeMgAK1z2eH1LGsNlWobPEjB47xNrNzOw4jVlHQ+j669GKfW3sSyDrxZiE4xkaqnVukZyi+hh9wCdawUYj+Vt2NWuKUC+num4Agg9Hl6nsSWsZFbrvnjI5jD1pLV0FSYrvZTQ2RxL5zo0ubNGnndAyNQpdl5aevvaPwBCi/qB8OdX0YgBegoiUUn+hrWc2pPGwiwQSn/BHW9iib2lXk3PLE4XqEHH6mR9CZxd1Myo/33bqzkwgi57w5PS5CH04NitiSqayLi/D+wAPUJ/GRvm57nHTIUXxd4SUqVHoupuG0td+zkxdyrRq1zc0qkqVUJVNtR01RlflZ3RKEZ2r3Noyw3F9Q+XZuGNe8tYEaflx6FylNaFudBy6X7irbv15X0ZiibV2nRu1bvpdTVST554CUEfXviUEgvml6uuvn4Mkv3VjWn8t5cIUk+ncizoYeCmJOr8GZAjdh1Mdm1FozAyNUv0bOo5rcjDUQ6wADzMz8wsUM/UAfy5U5xdQr10RUpJI66Hpi7S4kRzqWegAhc/6G8X1Mal2/EJDNRdceM6NCQj0ux0M48S51OzkYOBFR+kqSFJKT+jmWDmzh457c2gYE6WEQEiZGoVucm7NdP34vmGc1CIV0nb9riTTmnW+iR2m7Ne8toyjQh8rfjeqtOOH9McaGgnwL6+g49BVX1w3lInP96W1jFSdpxNS0UjVw67vu/40QADws4JMFSR9OfShYf31I8+DwXDYdQk3ZunoKAo9pjt41Wdubdit5zZskSmm4ly+0Sk6D3jarl87utAo32w044bq+i4+FVJnMME9IkJ0dEAIDl3NjavVMcwPKV60gomqizp+/gXdAa765zomQK2bqPa+T/ij6bCjU3485TeM9X4p/0PTvP58Co7pSkZMrF5X/4J9/bWUC1N0V74B6eLz2lDj+sPO22EeDP7ZaNmGMqBQV1PedDD4RoTo6kgD/tEKQ43y63SE1+FrQkj+Bajs/gV3P4Vp3fg70/tGhM4r9eAbNmvyL/gCCvW8uVSdroJk2o57PLvpsFOHlrOFaIkASvvaKnSWmOLQ+90U2bgiLR2yBghIS+v4E/lnLpKbqt2qGe9n8prQxMQp6r6IdSavL61gsoJ8Dl9zjLTfhjJRdb6cvu6QUu34Uy56C9Hdea2P+/anFey0lvs6lpjRjck3G9cAKFKw5Wl11BB6lPXVFySZksBayoUlpljrfBG7Kq2xXtn4Ky1ZU8RCiPRg8HRK1c1D2uIzoVlX5Tc0KGLfw25sQLM+c2xyQnpH7hioOm/H81iPrH0tRBN14xP+aEahvpadGSSln7u2E2v9UpR6/Dofjo+fzMTn9zwReg6Sqk7RNsoljEw2OHNjxklNEat2vCiXcVJzQgJZhACTKvFdfCPD3PjWrxiOE8zorBePMQHmOfahtRoPKQ+qpPh7ShSP7bNuTNaLH1Wnj3LxqUxoyuhVz87XQqyjWT/rZTSuR6cAhMJlGt8LkCpnb8tOU6Kh+LlrO6aM5zaxiClq01QVTkjT2UdpDUwHQ7eDYRw7tWE2D8N45Cf1K9xRiZaO8lRaJlohRfqeoXBMDtMUUpdHAHnMsU7Z+FovAyvlwlM2vsja5Ez3plwMNKYvhx5ib+YgSROKCfiNCTA7nn0OqpAyPQrdgCa8PfKGDeVjOqcZnnx6oonD9HeKGrIzfSiXAP4FXcExwI9yGZmyGD1TuE3rxp+eMK8bf/8CL7FoYKADKNYqEM4pWpUoszpc49mtFqInuKn5pSK/w85EuVDq+oeUqVLo3Y4oXa4LEByRsXnRuCq+OJGQsq4kVDveTlGjw86Tb2aWHjU6MwlOUVNEiDvSSq0cY2KRI5o1KT9vesJ02HmsG1s7PY/6KarPJmvVfY5N2a/+fLN23XiuP6OFGBAkjVzDFg1zQ7mMJKRMlUI3mWOAXwaYqR1XtDZBAbwwthxZG5CWq1NUxX3XD7sw1kvfIxtSxQGbohXc+2JOtQfcY+tNSMt33Zgcvf5hsyY0647QTf6FUDwxhZ7QjongwzFSfgGokqgj3OnQBkagdYoyxUYHAD5ZlTYu1O1hmxy0QOpsc1V+eShmtZyB7yFlURIAHyHN+GwoQwQG4Gs6K6VVflbqzHLNXB2O06SXmlM08nOKDg0Ou34knNPkTUkvgF/Y4oRG0mcG+1qI1cOX4sMxWR2Au/IbGK2gyJty0YfNUsCWoTjcpcyhCyFuE0IcFEIcEkJ8QPP5i4QQXxFCDIQQ/yJ8N5vFdnqrz11kZMjU81E2OULSbXCC0qo7pfxrcpiUBOCxoRqcoi5Ky76hPJxbBqUlRJqg5OzozWikatILhaqzWYguz0oddvooF/ewRZPVIYRILU3P9adL1it+7tIfUzkDwE35SSlTDl1rIRKiXLj+LSNCv8TDFoUQEYCPAHgTgJsBvFMIcXPla+cA/CKAfx+8h45iipH2XXwDw+LrR+5cqClyAgBmfJC+MQ7dnws1hcIBfhEhJitISrdFbPILAJ7xxIbDDvCr7dEEBHz6w11/tsPO5zISk38BUOvPY91oDjt/Lp4PKEyRWuo9/xBePtjSl9lWVseli9BvBXBISvmUlHII4A4Aby1+QUp5Skp5D4DRReijk5g84JMQK6bjL0uCcKEn7JSLD9+n3+AKhXKdUr73Hw5jPULqeWxwUxwwQItD1yfhuCu/gSVyQn3uIiNDyQifg8GULAX4XUZiitxR/fGzOvS+DmDiGHRpRzfHPoDC5BcA/JC1yS+l2nH1Sw1iQzmDy8Apeg2Aw4XXR7L3vEUI8R4hxAEhxIHTp09TmjBKI+Xig7SY1I3NrPNRxE3moY9T1GTGAx6ZohanaLG/1jYa5sY5ysCitLqRn/LTtTFDWDemQ6rY36Y2AL2y8cmqNEXuqPdcld/YMCbfCCBjOQOPJLBwh5SF8vNC+oZyBpc65QKgfkQDpN5KKT8qpdwvpdy/e/duShNGaXSKMhefz8FgRRPdjrPSMtVDB/wWn4nD7PpWfwzgp7BtqH43YidLAX7V98zrJpRT1GPd2A6pjvtlEEqhmyxW9/hxc+QO4EdH2SgXH6vX5N/i1qdR73mNSTO/l0O1xSMAri283gvg2MXpDl2Msa55ISt3RWFV6AGUlm/yggn5hdpQ3GgFH3qiyXpxDw1toFycrY44nDOd6RS1AYEQTlH1npe1anhO6nO3dvTrxucyEuuYosjbKaoLm/XJVDaBpMll55cuh34PgJuEEDcIIfoA3gHgzovbLX+xJR0AbghJSmlBbO7JFFYOncD3mTNOeSh0xkMRA/awxWJ/m9oA+E5R2waf6XUw8AgxndHWH1dRLu7t2NAsl6rzSm+3IHSf9WcMEAjkFFXz7vKsbI7eXlc4R+4MLFaQF+XSENXkk3sQUrpNX5BSjoUQ7wPwOQARgI9LKR8VQrw3+/x2IcRVAA4A2AogEUL8cwA3SykXL17XyzKME2yerQ/Hb0MpZM1bxDaqxCdBaX1k2ZjelEu9jdme+4bKDzsLh+4yN+pvqb9dFJpTtD7Hs90onzuXduxRLm6H5voowWyP5xRVVJxujmd7EdZHftFRpnG5rr/BKDY+p+LfaezPWF/OYKbnDihsilhF7kgpa07KWjsj8/qb8fBLNYEk1/UXWhoVOgBIKe8CcFflvdsLP59ASsVsmDQ5t5zigK1OyKj0HWtfbGGLXfewxUFGB+gWaepAdEcT8/36o57NEJLL4rONyefQVH/LpCh8xgQYNrgnQrdRJa6W3WAcW5G+i9JSKFRHsc32Olj3jtHnIf31cYIZzXOa+KV4eRCT9eeA0G1WR2GPVxPNqpL7FzSHb68r3IGAMXKsg25HOK+/0OJCuVwWYqIDJsqmeYKb+F31d5pkMLIpP/dstMEowaymDdU2txKgWtQuG8ruF3BHoWqhc+vlqNo9usPOC6E3UHVufoH0Dk8tQveJAMrLGejoqAhxIp242eE4QUfUa7yr/rgi68Eo1j6nTie9mNk1TX441jsQ1XwNXABFg18AcF1/GaAwXPPH9UsBypq6dDn0y0JMWYw9D4RuQ48+jj8breBDlQzGsRYhAX70xLrBdM45dA+ExHUg5jSSDiFF7hmng3GiHZNq2xUhmdZNfhmJx/PWIXQfZbNuWTdK+bmgdBMdoPrjqrTWLXPcc0y0G8cJxonUKlA/Dt3mFPVZfzGEMNChHv4FE8UGpM+qRehMGRqSFyacVvMEr+f8GldpWRR6FDknKNkWzUy344RsgFRR6PoihEjbcVESDgrdxTGVH3baDe5+aK6NYsvc8Dl0wJ2emAAB89y49Cfnd23Kz/Hw1VlSgKdT1IDQgezQdNlT2d+a65sRuhPlZ6WRshBTR4Q+2420lp0fHaXfU4Df+gst06PQTWGLGae17nBi2hCSj+mcb3Ammh2MY+3mBoC5XuQ0JtUfM5pwc7ZZNxSBQ9ch9NxJ67AZ1kd6zlq17To3pvoggEoCa27HhtDneu48se1g8EHoqdWhnxsfpWWzglxpLRu4mfFwyoezEGPt2gOyPeXoeF4bhll/oWW6FLoBTcw5clouSIuN0H0U+igxLr7ZXoS1oaNCH5rRxGyv44Yex+awMb8xmVGoUn5rDptqMEow1zcrG1frxbZuXJPAXA4pF0WxZlN+ngjdhKy9FLoFoc/1I6fntG553rMe1ovdKe/ueE79Uqa94DYm9bdCrL/QMhUK3RY/DqRIwOVBKQWp21A+ysaNi3ezGGwI3XXx2czDWUekr/7WvGYR+1AlCmHqlJ8yy10VhZFy8Ylysa2bbuTUjhWh930QevqdORuH7qj8zGPyCJsd2y07n72ge94zHoedNWzRg4tfH1sQuuMhBWTrz0ZHtQidLsM4QZxIbWgekCqKdQc0a3dKeSj0cYxeJGqV2ABg3hOFGhF63808jBOJUax3SgHpBndCj0OzslFKy3VM6d/l0RNrI/Nh5xPlYnIYq/74HeCaQyoQEMiVn8vha7HIfIDAwEJrzfXc1o3NWp31AQIWQDE5NPkIfX2UIHHyb1lAUtedugktU6HQ14fmjQC4L2IbHaAWkgvNsW5RNmrxrToeMFYO3YvDNCMtVyckkB4kur4AwNpw3Nwfy2E346X87DSSC0JKEon1UaI9pID0Wbk8JxtCz4HA0P1Z2Zz7rv6FuQYU6hJJ1ITQuVaHitn2AhSa9eezNwc2hJ5z+m70rD3KpaVcyLI6ShWJ7vQG/M1DLdLyQKHrI31Chm87dg6947Qx8w1l4fu4G1NZRi5Ky4aQcoTudGjaIoAijOLmSCKFdk1zM99381MMLOsm6qThj66HVL/bqV0VmLbtjtBXh2OLtRpBymalNc6sXjNC991TZgvRRfmpg9Xme1l1ARTW9ecWESeldIhyaRE6WWx0AODuQLQpP7UInBCbhd/NKZcACF3RKTZZs1gdQOaRd0B9axaFHnXSm3DUwWoTK4fpSWuZDs1Zx42pnqUJCDhbdhaErvrjemgaeVkPhL42MkenzPfc1rHydRhRqONh52IhOs9NT3/YeYEky/pzpVXTXAnzIdUidKbYogMAFeLnQStoNmanI9w3puX09qFcmqJcin029sXilALc0YRC3yY0O9eLnJD1YKQvhlVs24lKGsbGA9zVSdsEBOaclZZd+c05+jvWbZE7HiF+a8Ox8ZBSyL0JzSr60YbQnZ6TxS+Vtu8GKFYtz9uPDm1ef657yhpo0CJ0uqw5IC1XMx4wP6j5ftfdrDMhdC/qxuKU6ruFsdmcUun7bmii6dCcd+Sb18fmUDh1kLohdDu/CzQj9DWLRQa4jylH6FznqtWMd0+QWxvFZvrRMeomj0ayhgJ7RI5ZHJFOh9QoNtNIjlYH4IjQm6yXBqvD9ZC6GDIdCr2JJ+65c5iAfRG7OrfMTlHFNzcfDNaN6aj81Gax0VFOGX8WykW9v+q4wednTMrGM2zRQiMV+2zrC2AGAvP9rlNfVgZZOwEoP5viS7/jhma5lItan/MzeiXqvKcUdaPJFAXSQ9CJ8huaaUwfymV1GGOTBfgBDoddA43pekhdDJkKhb7qYjq78rIGp5Rqx53vsy+apg01HCcYxRKbDBsqnHnoVsVvbRgj6ghtDQzVHxcraHlgcdg5OkWllNY5ntAKbhy6zXR2UcTKajM9q1lnyi+cZWc+pNzW33J2SNmUn4tT3lauVvVnzcH3YkPo/aiDqCOcrOcV2/pzpPya9lSqJxLnKwNDylQo9MZIDlfKZRgb2wAyFOqwaFaHzRuqaWOuDDIl0YAmmhRO42HnOKa1UcphmupNu9ITq8MxNlsUn/pbNhmMEyTS/Ly3ZO0vD+zjmsQ26/sz309vl2qqcLg8SEMxrZnKjuvPBgQ6Alhet49pFKdAwAZugOY5Xh00HFKO0TJN62/zTLdxTGk7Y2MbQohsHTcDgZVhjE0mC9HR6lX7xbTH1fpecdhXoWUqFLqL0nIJ91oamJUN4I70l9bH2ss2gJTOEaJZEStlZDZ53dCEOhhM/dk808P6KGmsUbNmQcSqPy6Uy8ogNiqJnmNccj4mQztqrE2KomnduB6+q8OxcUyAx7oZjLFltqf9TAiRKr+GQ6qJfswVuuP629RkTTk8q5lux1hqdvNsF0sNYwIyysUGthys58E4Rc1mq9eN8lMUm2lPbXFcfxdDpkKhuzhFR7FsrGGxvN6g0B1N8GXLwaDQhCuyNrUziZaxL5qlBuWnFt9Kw6ZaHYy1FfOUzDtSLiuDsdHqANySeZYbxrTJESHZyhkA7k6y5cHYqPgAdytoeTDCZgN6BIAtsz0sNSgJWwIOAMz3/OgoE5p1pQ7TQ8o8N5v73ca1B2SUiwVQuFiIKw2H1OR52/uzPBgBaF5/TYfvxZCpUOhqsxhN8Fm3CbYpYkDxfQ5mnUM7TWg2R+jcMa27KfQmRbG0PsaWGT16BJRT1I3DtKHZrQ5KS31uREgzbmNSG9fG7wIOym9gNuOBdI6b+gKkyM80JiCjJzJlYuxLE7hxtDqWGygXZUm4rD/bXtg860q5uNChroeUfUzO689kITquv4shU6HQF9dTs84U4rd1Ln1Qi2v2zbAyMFMlQLoQmhbfYJwW9Le1M9ePco7SJOqQalLETWNaHowQZTH01nbW7e0srduR1vxMF6sD+4ZKEonVkZlyUf1p6otSIluaKJeGOV7MnqVpXIpbb0L6K5bMTCBdfy6bO1V+5kNz82wz5bKUzZ3p8M0PKcf1Z1Z+buuv6QDfPNPFyjBudCAurY+x1bb++s1W0IRGMvtehJisC5M0UX6uVu/FkKlQ6Evro1xp60SdvI1KqwFZb5vr4UKjArU/bNd2VnKErm9naz6mpsUXY/NM1+jMzJFWQzuL6yMjvwukY1pcH1mjHtKoCPOGAhRCb5jjBoTu6kBcXB9hptsxIvStc0ppNW9w2/PeOpuGP9oov8E4xjBO7PTETDePPjGJ6qtpP8z3I3Q7wmEd20Mxc5DE3FO58rMo4ySRWGzY41vneo3PqemQ6nQENve7DiDJ3o46lFvKhSiLa/bTe6sjrbDcgEK3zfWwMoytDsQmikO147qhbBEh/W7HCVlbTV5H83BpfZwrOJ1sm+thFEur2bvSsBGAVIk2KtAG68XVgbi4ZlcS27LPXA5xE8UBFE15cztq3dgOuzQixN4XtR5Mz0oI4Qwo5vuRMYR366zbYde0p9QztB2+y8MxpJyAGJ24jEmtcRs95mJNLQ3G6Hc7xqgmV6f8xZDpUOgOpzfgQk/YnVvbHNppOr1VO02Lb2F1mH533jKu2WZUkjramhHSUgM3u7g+sm6o7Q7K7/xq+tn2pjE5HFKAGaEDqRJtVuh2ILB9vg8AuLA2tLazsDrCjuy7OsmRvk1p5dFIFsrF8ZACJmtVJ9vmelhoWH/nV4f5M9WJ2lNN1tRKQwSQCz3mMqbtc718z5hkIV9/5mflRPmtj410H5A6egE4Re+ElulQ6Gt2ZePCEw/HCVaHMRux5YvG2k6/UaGfXx0i6ohGy6Np8V1YG1mRtQvlkiQSyw3RCmpu1Ph1cj7bcHbl14yQ1NzZnvmmmciJcuE+byklFlZH2L7JfkgBdiAwGRPPgZgjdBuane81gpuF1VGj4kv/XvOzsvUlR+hWha5oJPv6WxqMrVy86/prmpsLDZadsgBahE6UxfWxE0K3KQr1sK/YZH7YudKyPPCzKwMAwM7N9nYurNn55vOrI2yf6xm5bwDY4qD8zq0MrWNy2ZjK5LUq9HmXwy6dYztC72JpfWS9ZODcyhDz/cgaF79trpc/U5M0AYFNGd9sO6RWhyn3bVMSLtETZ1fSvjatm5VhbE0rv7CWOsFtFNC2uZ51TED6rHZYDqmZboSZbseq/EZxgoXVkXVMW3MgYH5WbodUH1LaLYbzqyMIYUf6qYXI21PdqIPNM10sNFh2F0OmQqGfW7Gbh5v7XXQErBv8zHKqiHc5LD6b0jq3og6GGeN3FN9sCx1bWB1aFZ9qp8nMPLs8xM7N5r7MdDuY70d5v01tAMAuSzsuaFZRLk0IKZF2c/Xs8sC6oQBg95aZ/Jna+mPb3C588wT1NSN92/o7t9y8btT8q+ehk4VsTDYgsN2J8rMjdKD5YFDj3Wl5VruzMZ1Zso0p/YxvPQ+xdbanvVxFyda5Li647KmG9bdrcx9nLM/pYsllr9DXRzEurI2wZ4t5I3Q6Aru3zODUonmDuyhitWnPW5TfuZUhOsJOuShFfd62GVZGjUprT8OYkkTi/Kp98Qkh0naWzO2czj7bbZljtfltB4yLybtn6ywA4NTiuvE7Z1fshxSQKj/bhpJS4tTSunXdAKnlYVNaLrys+hu2OT7ngNDV/NsOqlNLg1xJmmT7fL/Rejm/OrQeUgCwZ+sMTi2Zn5PLnlJjOt0wJvX3TLJ9zmFPrY6ax7RlFqeXB1YLMV1/zYDitGVuLpZc9gr9tMPDBoCrts7ipMOGsinRq7alyub4Bbuy2THfN0YHqL4AwAlLO2ccUOiVW2dwenlg5A0X1kZIpH1MQLqIbQrURaHv3jwDIYATlnbOLA0x14usCSJXZn/jZMPh24SQdm+ewYW1kZGeWBqMsT5KGtfNlVtm7WPKFJGtP9vne+h3O42HVC8SVmebsh5PW9bxqaVB45j2bJ3B0vrYGCc9HCdYWBthp0URA2puLM9puXlPzfYibJnt2se0OEBHwNqfK7M9ddK6/gaNQOCqrTMYxdJ44CmQ5GIh2sZ0seSyV+j56b1l1vq9PVvtSkstBBu6me93sW2uh+MX1sz9WVy3Kj4AeN72OQDAsQV9O1JKHF1Yy79nkqu2ziJOJM4a0E0+pob+7N5qR+hKadnmpt/tYPfmGeOYgHS81+ywj0ltTJsSPbU0sFJjALBri52eUJaN+nsmed72ORy3jOlo9pltXEIIXLl1pmFM69i1ecZKleRo1qr81hv3wjXZujKt4xMX1iHl5HsmuXJbw55aclx/m2caEHq6p2xUyfO2p2O2rr8LzXuqaf2dXRkiTmSjFbR7c6vQSaKUViPS2jpjPb0Pn1vD1tmuNUwQAK7eNmtF1ofPrWHvjnlrG02Lb2F1hNVh3Lih9uSoRL9wDp9bBQBc29CfK7fM4uTiutFJe2xhDf1ux0qVAJnys8yNyyGlnqPpWa0NY5xeGjSPKWvHpLRcD7trtqcI3VRx8ej5NXQ7olGJXrV11rr+jpxbw7VX2Me0K7OCTHMcJxKnlwb52E1y9bb0GRxd0LfjckgB6bo5uzI0WkGHz6Xt7G1oZ8/WGeuheWJx0Di/V2zqY6bbMe6pJJE4vrCe7z2TXLlNUX6GPXU+21MNz2rP1lksWqygiyWXvUJ/+swKAGDfzk3W7127Yx7nV0dGjvfw+dXGhwSkSuvIeTOyTtuxL+Atsz1sme3mG6cq+YZqUH5KqT19dkX7+eGsn9c1jGvfrnmsDmPjwfDUmRXs2zlvpZFUf48a5gZIx9U0pvl+Fzs39fHMGf2YjmQb6rqd9jE9f/dmAMChU8vaz586nb5/wy77url6+xwSaUZsR86v4erts1b0CAB7d8zjmTOrxs+fO7faeEjN9iJcu2Meh07rx3T43CrGicS+hjEppWZ6VmqOmw5fpagVcKjKc+dWceXWGWs0EpA+qydPrxgBxVOnlxvHJIRI159hT51aGmAYJ9jrOKZnTHvqnJtCf/7utL9PGp7VxZLLXqEfOrWMq7bOWpMXAODFV28FADx2fFH7+cETS7kSsMkLrtyCJ08va1HJkfNrWB3GTu3cuGcznji+pP3s8ayPN11pb+fGPZvRi0T+/aocPLGIHfO9xmiZF12Vzs0TJ/TtPHlqufHAVP155uyKtqbGycV1nFsZ4qY9zXNz8/O24nFDXw6eTOfsO3bZ29m7Yx6zvQ6+eVK/oQ6dWsbmmW7uzzCJ6u/jhmf1xIlF3LRni7UNAHjx1VtwIpuDqiysDnFicR3P39M8xzft2YxDljEB6XOwydXb5rCpHxmf9xMnljDT7eDaBmR98/PSdfPoMX073zjptqdu2rMZF9ZGWopibRjj6MIabnRo5/l7NlufEwDc2PCsdm+ewa7NfTxmGNPBE0vodkQjSLrpyvTvmNbfxRInhS6EuE0IcVAIcUgI8QHN50II8RvZ5w8JIV4Rvqt6OfDsObxs77bG76nF99CRC7XPTlxYx/EL607tvGzvNoxiqV3E9z13Pv9Ok7x873Y8fPSCtr7HA4cXMN+PcEOD0up3O7hxzxY8dGRB+/l9zy3gZXu3W3lZAHjhVenie/BwfW7OLg/w1JkV3HLddmsbAPDya7chkfo5vv+5tI/f6fKsrt6Kb5xY1patve/ZBfS7nbzPJok6Ai+8cgvuP7yg/fzAs+fx0mu2Ns7NS563DVFH4P7s2RZlcX2EQ6eW8dJrXMaUfkf3rFQfX753e2M7L7o6BRS68Lx7nj2HXiTwIoe5eek12/JnUuvPc+fx4qu3omuoYa7kxj2b0Y862ue9Nozx+PFFvMxhTAps6Z7Vfc+dh5TAd+7d2tjOLddux9NnVrRRaGqsL7nG3o4QAi++eqt2TKqdF161pdHquP6Kecz1IjxgWH8XSxoVuhAiAvARAG8CcDOAdwohbq587U0Absr+vQfAfw3cT60ceOYcDp9bw+tfsLvxu7s2z+Cl12zFn91/tFaL5ZP3HQEAvPGFze28+jt2ohcJfPLeI6X3pZT41H1HsWvzDF7yvOYN/oYX7MbaKMbdjxwvvb88GONzj57E62/a3WjGqz5/9alzNbP3oSMLOHRqGW9wmJttcz288voduPPBo7UD5tP3HwUAvP6m5nb277sCM90O/uSAbm6OYNtcD7dcu72xnTe8cDeGcYLPPnSs9P7aMMZdDx/Ha5+/01hHoyhvfOEe3Pfc+ZxeUXLwxBIePbbotG7m+hFu3XcFPvvQsdoFCp954BgSCac5fuX1O7B5potP3ne09tmn7juKzTNdvPL6HY3tfO+L9mCcSNz5YHlu1kfp3Oy//gpr5Uclr3/Bbjx89AIOnigj2qdOL+P+wwtOe6EXdfCaG3firoeP1w7fzz54DKNY4vUv2NXYziuu34Gts1186r4jNdrl0/cfxUy3g1tv2Nk8pmyNqv2sZDhO19L+63dYk5OUvPGFe3Dw5BIerij1w+dW8fVnzjk9727UwetfsAt3P3KilgV7bGGtMUeCKqLpTkAhxKsB/D9Syh/IXn8QAKSU/7bwnd8G8EUp5R9lrw8CeKOU8rimSQDA/v375YEDB7w7/MWDp/Brf/4YVgYxziwPsHNzH1/4P99orVei5DMPHMUv3fEAdm7qY9tcDxLAOElw+NwavvdFe/Dxn3mVUx/+1Z89jD/46nO4ZvscZrodSKSK+PTSAL9y24vwT9/4/MY2xnGCH/wv/4BvnlrGtTvmIISAlBJnV4ZYHozxJ//k1di/74rGdg6fW8VtH/47JDJ12EqkCvTYwjq2znXx+fe/oTFBBAD+4pETeO8f3IsrNvXTmF4BDEYJji6s4bU37sQfvvu7HWYG+H8/+xg+/qWn87lJZFqw69TSAP/H970Av/R9NzW2kSQSP/xbX8IjRy/g2ivm0clQ9NnlARbXx/jv7/4uvObGZkVxcnEd3/cf/hbjROKqbbMQSCs+nlkeYNNMF194/xsaw9gA4G8OnsLPfuIe7JjvYcd8v7RuXrVvB/74n7y6EekDwIf+4gn81y8+iau3zWKuF0EiPaROLK7jF974fPzybS9qbCNJJH7st7+Ce587j+uumM9ueEqwsDrEyjDGJ372VfhHL9zT2M7ppQG+/z/9be58lwASKXFycR29qIPPv/8NjRFAAPClQ2fwro99Ddvmeti5KZ2bOEl9SS+7Zhs+/QuvbfS9AMCHP/8NfPjz38RVW2cxPxNhHKflJs6tDPHu77kB/+oHqxiyLlJKvOtjX8OXnzyL63fOIxICiZRYWBthYXWE3/7JV+IHXnJVYzvnV4b4x//pb7G4Pi75fI5fWEMkBP7y/W9o9AUBwD3PnMOP/fZXsGWmi12bZyCRVnw8vTTAT79mH371h17S2IZOhBD3Sin3az9zUOg/CuA2KeW7s9c/CeC7pJTvK3znzwH8upTyH7LXXwDwK1LKA5W23oMUweO666575bPPPus9mPueO4/f/funsWkmwo5NffzEd13v5MxUcvfDx/HXT5zKL+3tiJQX/5nX7Gvk4ZUMxwl+/yvP4JGjFxBn09eLBF79HTvxo6/c67S5gXRTfeJLT+fOy45InYI/9PKr8ZrnNyssJY8dW8Qd9zyHcytDCCEgkEZv/MR3X9/o9CvKXz56An/12Mm0zC2AmaiDG3Ztwk+/dp8TsgHSdO8/+OqzeOjIBYwTCQGg2xG49YYr8GP7r3Xa3EAaa/6JLz2NZ86mlodAWiPjzd95NV7nYC0oeeLEIv7oa8/h3OoIiZSY70XYNtfDO269rpFrLsrfPHEKdz9yHKvDGB0hIETqiP+5197QGBmlZBwn+KOvP4f7n1vAKMsd6HUEXnH9Drzz1uucLDIAuLA6wu/+w1N45uwq4kRiptfBpn4X3/viPU7KXMmhU0v4g68+hzPLg3xMO+b7+PHvug4vuLLZL6Dkb79xGnc9dDy/tEUA2LdrE372NfuwoyFeW0mcSPyPew7jwDPnMIgT9DoC8zNdvPDKLfjx77rOeIVdVS6sjfCJLz2Np06vQCLdUzPdDr7/5qvwfTdf6Tymp8+s4P/7yrM4tbSe76krNvXx9lddm1NELvKlQ2fw2QePYXkwhhACm/oRdm+ZwdtfdW1jNJxJuAr9bQB+oKLQb5VS/rPCd/4ngH9bUei/LKW819QuFaG30korrXw7i02huxx7RwBcW3i9F8AxwndaaaWVVlq5iOKi0O8BcJMQ4gYhRB/AOwDcWfnOnQB+Kot2+W4AF2z8eSuttNJKK+GlkTSWUo6FEO8D8DkAEYCPSykfFUK8N/v8dgB3AXgzgEMAVgH87MXrciuttNJKKzpx8gJKKe9CqrSL791e+FkC+N/Ddq2VVlpppRUfuewzRVtppZVWWkmlVeittNJKK1MirUJvpZVWWpkSaRV6K6200sqUSGNi0UX7w0KcBuCfKprKLgBnAnbncpB2zN8e0o7520M4Y75eSqlNl94whc4RIcQBU6bUtEo75m8Pacf87SEXa8wt5dJKK620MiXSKvRWWmmllSmRy1Whf3SjO7AB0o7520PaMX97yEUZ82XJobfSSiuttFKXyxWht9JKK620UpFWobfSSiutTIlcdgq96cLqy0mEEB8XQpwSQjxSeO8KIcRfCSG+mf2/o/DZB7NxHxRC/EDh/VcKIR7OPvsN4Xpl0rdYhBDXCiH+RgjxuBDiUSHEL2XvT/OYZ4UQXxdCPJiN+d9k70/tmJUIISIhxP3ZjWZTP2YhxDNZXx8QQhzI3vvWjllKedn8Q1q+90kA3wGgD+BBADdvdL8Y43k9gFcAeKTw3r8D8IHs5w8A+FD2883ZeGcA3JDNQ5R99nUAr0Z6+9fdAN600WMzjPdqAK/Ift4C4BvZuKZ5zALA5uznHoCvAfjuaR5zYezvB/DfAfz5tK/trK/PANhVee9bOubLDaHfCuCQlPIpKeUQwB0A3rrBfSKLlPLvAJyrvP1WAP8t+/m/Afjhwvt3SCkHUsqnkdaev1UIcTWArVLKr8h0Nfx+4XcuKZFSHpdS3pf9vATgcQDXYLrHLKWUy9nLXvZPYorHDABCiL0A/lcAHyu8PdVjNsi3dMyXm0K/BsDhwusj2XvTJFfK7Lan7H91669p7NdkP1ffv6RFCLEPwP+CFLFO9Zgz6uEBAKcA/JWUcurHDODDAH4ZQFJ4b9rHLAH8pRDiXiHEe7L3vqVjdrvm/tIRHZf07RJ3aRr7ZTcnQojNAD4J4J9LKRctFOFUjFlKGQO4RQixHcCnhRAvtXz9sh+zEOIHAZySUt4rhHijy69o3rusxpzJa6WUx4QQewD8lRDiCct3L8qYLzeE/u1wGfXJzOxC9v+p7H3T2I9kP1ffvyRFCNFDqsz/UEr5qeztqR6zEinlAoAvArgN0z3m1wJ4ixDiGaS06PcKIf4A0z1mSCmPZf+fAvBppBTxt3TMl5tCd7mw+nKXOwH8dPbzTwP4TOH9dwghZoQQNwC4CcDXMzNuSQjx3Zk3/KcKv3NJSda/3wXwuJTyPxY+muYx786QOYQQcwC+D8ATmOIxSyk/KKXcK6Xch3SP/rWU8icwxWMWQmwSQmxRPwP4fgCP4Fs95o32DBM8yW9GGh3xJIB/udH9YY7ljwAcBzBCejL/PICdAL4A4JvZ/1cUvv8vs3EfRMHzDWB/tnieBPCbyDKAL7V/AL4Hqfn4EIAHsn9vnvIxvwzA/dmYHwHwr7P3p3bMlfG/EZMol6kdM9LIuwezf48q3fStHnOb+t9KK620MiVyuVEurbTSSiutGKRV6K200korUyKtQm+llVZamRJpFXorrbTSypRIq9BbaaWVVqZEWoXeSiuttDIl0ir0VlpppZUpkf8fs1/QIeVtp1sAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.integrate import solve_ivp \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "y0=[0,1]\n",
    "k,p = 1,6\n",
    "t0,tf = 0,50\n",
    "#sol = solve_ivp(f1,(0,10),y0,args=(1,0.1),t_eval=np.arange(0,10,0.01))\n",
    "sol = solve_ivp(f2,(t0,tf),y0,args=(k,p),t_eval=np.arange(t0,tf,0.01),atol=10**(-16),rtol=10**(-5))\n",
    "KE = 1/2 * sol.y[1]**2\n",
    "PE = Vr2(sol.y[0],k,p)\n",
    "\n",
    "Etol = KE + PE\n",
    "\n",
    "plt.plot(KE,label=\"KE\")\n",
    "plt.legend()\n",
    "plt.show()\n",
    "plt.plot(PE,label=\"PE\")\n",
    "plt.legend()\n",
    "plt.show()\n",
    "plt.ylim(0.49,0.51)\n",
    "plt.plot(Etol,label=\"Etol\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.1.2 量子系统"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "量子力学描述了在原子或亚原子尺度上发生的现象x。这是一种统计理论，其中粒子位于点x周围的区域dx的概率为$P=|\\psi(x)|^2 dx$，其中$\\psi(x)$被称为波函数。如果一个在一维运动的能量为E的粒子遇到势$V(x)$，其波函数由一个常微分方程（或一个以上维度的偏微分方程）确定，该方程被称为与时间无关的薛定谔方程：\n",
    "$$\n",
    "\\frac{-\\hbar^2}{2 m} \\frac{d^2 \\psi(x)}{d x^2}+V(x) \\psi(x)=E \\psi(x) \\tag{1}\n",
    "$$\n",
    "尽管说我们求解能量E，但在实践中，是求解波向量κ。束缚态的能量是负的，所以我们用\n",
    "$$\n",
    "\\kappa^2=-\\frac{2 m}{\\hbar^2} E=\\frac{2 m}{\\hbar^2}|E|.\n",
    "$$联系两者。\n",
    "则薛定谔方程的形式\n",
    "$$\n",
    "\\frac{d^2 \\psi(x)}{d x^2}-\\frac{2 m}{\\hbar^2} V(x) \\psi(x)=\\kappa^2 \\psi(x)\n",
    "$$\n",
    "当问题表明粒子是有界的，即它被限制在空间的某个有限区域。使$\\Psi(x)$具有有限积分的唯一方法是使其指数衰$x\\to \\pm \\infty$（其中势消失）：\n",
    "$$\n",
    "\\psi(x) \\rightarrow \\begin{cases}e^{-\\kappa x}, & \\text { for } x \\rightarrow+\\infty \\\\ e^{+\\kappa x}, & \\text { for } x \\rightarrow-\\infty\\end{cases} \\tag{2}\n",
    "$$\n",
    "总之，尽管我们通过ODE求解公式(1)很简单，但我们也必须要求解$\\Psi(x)$同时满足边界条件(2)。这个额外的条件将ODE问题变成了一个特征值问题，该问题仅对能量E的某些值具有解（特征值）。基态能量对应于最小（最负）的特征值。基态波函数（本征函数），我们必须确定它才能找到它的能量，它必须是关于x=0的无节点甚至（对称）的。激发态具有更高（负能量较小）的能量和可能是奇（反对称）的波函数。\n",
    "\n",
    "\n",
    "我们将使用一个简单的模型，其中势V(x)是一个有限平方阱：\n",
    "$$\n",
    "V(x)= \\begin{cases}-V_0=-83 \\mathrm{MeV}, & \\text { for }|x| \\leq a=2 \\mathrm{fm} \\\\ 0, & \\text { for }|x|>a=2 \\mathrm{fm}\\end{cases}\n",
    "$$\n",
    "其中深度值为83MeV，半径值为2fm是核的典型值。有了这个势，薛定谔方程就变成了\n",
    "$$\n",
    "\\begin{aligned}\n",
    "& \\frac{d^2 \\psi(x)}{d x^2}+\\left(\\frac{2 m}{\\hbar^2} V_0-\\kappa^2\\right) \\psi(x)=0, \\quad \\text { for }|x| \\leq a, \\\\\n",
    "& \\frac{d^2 \\psi(x)}{d x^2}-\\kappa^2 \\psi(x)=0, \\quad \\text { for }|x|>a . \\\\\n",
    "&\n",
    "\\end{aligned}\n",
    "$$\n",
    "为了给出这里的常数比，我们在分子和分母中插入$c^2$：\n",
    "$$\n",
    "\\frac{2 m}{\\hbar^2}=\\frac{2 m c^2}{(\\hbar c)^2} \\simeq \\frac{2 \\times 940 \\mathrm{MeV}}{(197.32 \\mathrm{MeV} \\mathrm{fm})^2}=0.0483 \\mathrm{MeV}^{-1} \\mathrm{fm}^{-2}\n",
    "$$\n",
    "\n",
    "特征值问题的求解将常微分方程的数值解与满足边界条件的波函数的试错搜索相结合。这分几个步骤完成：\n",
    "1. 从最左边的$x=-X_{\\max } \\simeq-\\infty$开始，其中$X_{max}\\gg a$。由于该区域的势V=0，这里的解析解为$e^{\\pm \\kappa x}$。因此，假设那里的波函数满足左手边条件：$$\\psi_L\\left(x=-X_{\\max }\\right)=e^{+\\kappa x}=e^{-\\kappa X_{\\max }}$$\n",
    "2. 使用ODE求解算法将$\\psi_L(x)$从$x=-X_{max}$向原点（向右）逐步推进，直到达到匹配的半径$x_{match}$。这个匹配半径的确切值并不重要，我们的最终解决方案应该与之无关。\n",
    "3. 从最右边开始，也就是说，在$x=+X_{\\max } \\simeq +\\infty$处，使用满足右边边界条件的波函数：$$ \\psi_R\\left(x=+\\kappa X_{\\max }\\right)=e^{-\\kappa x}=e^{-\\kappa X_{\\max }} $$\n",
    "4. 使用ODE求解算法将$\\psi_L(x)$从$x=+X_{max}$向原点（向右）逐步推进，直到达到匹配的半径$x_{match}$。 这意味着我们已经跨过了势阱。\n",
    "5. 为了使概率和流在$x=x_match$处连续，$\\psi(x)$和$\\psi^\\prime(x)$必须在那里连续。要求比率$\\psi^\\prime(x)/\\psi(x)$（称为对数导数）是连续的，将两个连续性条件封装为一个单独的条件，并且独立于$\\psi$的归一化。\n",
    "6. 我们需要一个能量的起始值来使用我们的ODE求解器。在这种情况下，我们从对能量的猜测开始求解。基态能量的一个很好的猜测是一个比井底能量稍高的值，$E>-V_0$。\n",
    "7. 因为任何猜测都不太可能是正确的，所以左波函数和右波函数在$x=x_{match}$时不会完全匹配。这是没什么问题，因为我们可以使用不匹配的量来改进下一个猜测。我们通过计算差值来测量左右波函数的匹配程度\n",
    "$$\n",
    "\\Delta(E, x)=\\left.\\frac{\\psi_L^{\\prime}(x) / \\psi_L(x)-\\psi_R^{\\prime}(x) / \\psi_R(x)}{\\psi_L^{\\prime}(x) / \\psi_L(x)+\\psi_R^{\\prime}(x) / \\psi_R(x)}\\right|_{x=x_{\\text {match }}},\n",
    "$$\n",
    "其中分母用于避免过大或过小的数字。接下来，我们尝试一种不同的能量，注意$\\Delta(E)$变化了多少，并用它来推断下一种能量的猜测。搜索继续进行，直到左波和右波$\\psi^\\prime/\\psi$在一定公差内匹配。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8683421972946235\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "-1.2334197224689195e-11"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import solve_ivp \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import root,bisect\n",
    "\n",
    "def f1(x,y,k):\n",
    "    if np.abs(x) > 2:\n",
    "        return [y[1], k**2 * y[0]]\n",
    "    else:\n",
    "        return [y[1],(k**2 - 0.0483*83)*y[0]]\n",
    "\n",
    "def diff_f(k,plot=False):\n",
    "    x_max = 10\n",
    "    yl0 = [np.exp(-k*x_max),k*np.exp(-k*x_max)]  \n",
    "    yr0 = [np.exp(-k*x_max),-k*np.exp(-k*x_max)]\n",
    "    x_match = 1\n",
    "    sol_l = solve_ivp(f1,(-x_max,x_match),yl0,rtol=10**(-6),args=(k,),t_eval=np.arange(-x_max,x_match,0.01))\n",
    "    sol_r = solve_ivp(f1,(x_max,x_match),yr0,rtol=10**(-6),args=(k,),t_eval=np.arange(x_max,x_match,-0.01))\n",
    "    if plot:\n",
    "        plt.plot(np.arange(-x_max,x_match,0.01),sol_l.y[0],label=\"yl\")\n",
    "        plt.legend()\n",
    "        plt.show()\n",
    "        plt.plot(np.arange(x_max,x_match,-0.01),sol_r.y[0],label=\"yr\")\n",
    "        plt.legend()\n",
    "        plt.show()\n",
    "        plt.plot(np.arange(-x_max,x_max,0.01),np.concatenate([sol_l.y[0],sol_r.y[0][::-1]]),label=\"yr\")\n",
    "        plt.legend()\n",
    "        plt.show()\n",
    "    diff = (sol_l.y[1][-1]/sol_l.y[0][-1] - sol_r.y[1][-1]/sol_r.y[0][-1])/(sol_l.y[1][-1]/sol_l.y[0][-1] + sol_r.y[1][-1]/sol_r.y[0][-1])\n",
    "    return diff \n",
    "#print([diff_f(i) for i in np.arange(0,2,0.1)])\n",
    "eig = bisect(diff_f,0,1)\n",
    "print(eig)\n",
    "diff_f(eig,True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "探索\n",
    "\n",
    "1. 通过使用任意的起始能量值来检查搜索过程的工作情况。例如，因为没有束缚态能量可以位于井底以下，所以尝试$E\\geq-V_0$，以及$V_0$的一些任意分数。在任何情况下，都要检查得到的波函数，并检查它是否对称和连续。\n",
    "2. 逐渐增加势能的深度，直到找到几个束缚态。观察每种情况下的波函数，并将波函数中节点的数量与阱中束缚态的位置相关联。\n",
    "3. 探索当你改变井的深度$V_0$时，束缚态能量是如何变化的。特别是，当你不断减少深度时，观察本征能量向$E=0$移动，看看你是否能找到束缚态具有$E\\simeq 0$的潜在深度。\n",
    "4. 对于固定的阱深$V_0$，探索束缚态的能量如何随着阱半径$a$的变化而变化。\n",
    "5. 进行一些探索，发现$\\left( V_0，a \\right)$的不同组合具有相同的基态能量（离散模糊性）。几种不同组合的存在意味着基态能量的知识不足以确定井的特征深度。\n",
    "6. 修改求解奇波函数的本征值和本征函数的程序。\n",
    "7. 求解线性势的波函数：$$V(x)=-V_0\\begin{cases}|x|，&& \\text{for}|x |＜a，\\\\0，&&\\text{for} |x |＞a \\end{cases}$$\n",
    "这里的势比方形阱的势小，所以你可能会期望更小的结合能和更小的受限波函数。\n",
    "8. Newton-Raphson扩展：使用Newton-Rephson方法代替二分算法来扩展特征值搜索。确定它的速度和精度。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.1.3 散射问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "问题： 弹球从障碍物上的经典散射是一个连续的过程。然而，在弹球机上进行的实验中观察到，在某些条件下，弹球会经历多次内部散射，最终形成与初始轨迹明显无关的最终轨迹。问题是确定这个过程是否可以被建模为来自静态势的散射，或者弹球机中是否必须内置导致混沌散射的主动机制。\n",
    "\n",
    "建模：弹球机中弹球撞击缓冲器的模型是点粒子从静止的二维势场中散射出来的：\n",
    "$$\n",
    "V（x，y）=±x^2 y^2 e^{-（x^2+y^2）}。\n",
    "$$\n",
    "该势在xy平面中有四个圆对称峰。这两个符号分别对应排斥势和吸引势（弹球机只包含排斥相互作用）。由于该势中有四个峰，我们怀疑可能会有多次散射，其中射弹在峰之间来回反弹，有点像弹球机。\n",
    "这个问题的理论是经典动力学。想象一个散射实验，其中弹丸以确定的速率v和碰撞参数b从距离目标无限远的地方开始，入射到目标上。在与目标相互作用并移动到距离目标几乎无限远的地方后，散射粒子以散射角θ被观察到。由于势能不能反冲，弹丸的速度不变，但方向会改变。实验通常测量散射粒子的数量，然后将其转化为函数，即微分截面$\\sigma(\\theta)$，它与实验装置的细节无关：\n",
    "$$\n",
    "\\sigma(\\theta)=\\lim \\frac{N_{\\text {scatt }}(\\theta) / \\Delta \\Omega}{N_{\\text {in }} / \\Delta A_{\\text {in }}} .\n",
    "$$\n",
    "其中，$N_{\\text{scatt }}(\\theta)$是在角度为θ的立体角ΔΩ内每单位时间散射到探测器中的粒子数，$N_{in}$是在横截面积为$\\Delta A_{in}$的靶子上每单位时间入射的粒子数，式中的极限是针对无穷小的探测器和面积。\n",
    "\n",
    "![Alt text](pic/sec3-1-scattering.png)\n",
    "\n",
    "散射截面的定义是实验学家用来将他们的测量值转换为理论可以计算的函数的定义。作为理论家，我们求解从势散射的粒子的轨迹，并从中推断出散射角θ。一旦我们得到散射角，我们就可以根据散射角对经典碰撞参数b的依赖关系来预测微分截面：\n",
    "$$\n",
    "\\sigma(\\theta)=\\left|\\frac{d \\theta}{d b}\\right| \\frac{b}{\\sin \\theta(b)} .\n",
    "$$\n",
    "在模拟中，你应该会发现一个惊喜，对于某些参数，$d \\theta / d b$ 可以变得非常大或不连续，这相应地导致大而断续的散射截面。\n",
    "要解的动力学方程就是牛顿定律，它适用于具有上面势能的x和y方向的运动：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathbf{F} & =m \\mathbf{a} \\\\\n",
    "-\\frac{\\partial V}{\\partial x} \\hat{i}-\\frac{\\partial V}{\\partial y} \\hat{j} & =m \\frac{d^2 \\mathbf{x}}{d t^2}, \\\\\n",
    "\\mp 2 x y e^{-\\left(x^2+y^2\\right)}\\left[y\\left(1-x^2\\right) \\hat{i}+x\\left(1-y^2\\right) \\hat{j}\\right] & =m \\frac{d^2 x}{d t^2} \\hat{i}+m \\frac{d^2 y}{d t^2} \\hat{j} .\n",
    "\\end{aligned}\n",
    "$$\n",
    "$x$ 和 $y$ 运动的方程是同时的二阶 ODE：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "m \\frac{d^2 x}{d t^2}=\\mp 2 y^2 x\\left(1-x^2\\right) e^{-\\left(x^2+y^2\\right)}, \\\\\n",
    "m \\frac{d^2 y}{d t^2}=\\mp 2 x^2 y\\left(1-y^2\\right) e^{-\\left(x^2+y^2\\right)}。\n",
    "\\end{aligned}\n",
    "$$\n",
    "由于力在峰值处消失，这些方程告诉我们峰值在$x= \\pm 1$和$y= \\pm 1$处。将这些值代入势能得到$V_{\\max }= \\pm e^{-2}$,这为问题设定了能量尺度。\n",
    "\n",
    "我们使用与单次二阶ODE相同的算法，只是现在数组将是4D而不是2D：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{d \\mathbf{y}(t)}{d t} & =\\mathbf{f}(t, \\mathbf{y}), \\\\\n",
    "y^{(0)} & \\stackrel{\\text { def }}{=} x(t), \\quad y^{(1)} \\stackrel{\\text { def }}{=} y(t), \\\\\n",
    "y^{(2)} & \\stackrel{\\text { def }}{=} \\frac{d x}{d t}, \\quad y^{(3)} \\stackrel{\\text { def }}{=} \\frac{d y}{d t},\n",
    "\\end{aligned}\n",
    "$$\n",
    "其中，$y^{(i)}$的赋值顺序是任意的。根据这些定义和方程，我们可以为力的函数赋值：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "f^{(0)}=y^{(2)}, &\\quad f^{(1)} = y^{(3)}\\\\\n",
    "f^{(2)}=\\frac{\\mp 1}{m} 2 y^2 x\\left(1-x^2\\right) e^{-\\left(x^2+y^2\\right)}&=\\frac{\\mp 1}{m} 2 y^{(1)^2} y^{(0)}\\left(1-y^{(0)^2}\\right) e^{-\\left(y^{(0)^2}+y^{(1)^2}\\right)}, \\\\\n",
    "f^{(3)}=\\frac{\\mp 1}{m} 2 x^2 y\\left(1-y^2\\right) e^{-\\left(x^2+y^2\\right)}&=\\frac{\\mp 1}{m} 2 y^{(0)^2} y^{(1)}\\left(1-y^{(1)^2}\\right) e^{-\\left(y^{(0)^2}+y^{(1)^2}\\right)}。\n",
    "\\end{aligned}\n",
    "$$\n",
    "为了从我们的模拟中推断出散射角，我们需要检查散射粒子在与目标“无限”分离时的轨迹。为了近似，我们等到散射粒子不再感受到势能（假设为$|\\mathrm{PE}| / \\mathrm{KE} \\leq 10^{-10}$）并称之为无穷大。然后，散射角可以从速度分量中推断出来，\n",
    "$$\n",
    "θ= tan ^{-1}\\left(\\frac{v_y}{v_x}\\right)=atan2\\left(\\mathbf{v}_{\\mathbf{x}}, \\mathbf{v}_{\\mathbf{y}}\\right) .\n",
    "$$\n",
    "\n",
    "1. 应用ODE方法求解具有4-D力函数的二阶常微分方程组。\n",
    "2. 初始条件是 (a) 只有速度的 $y$ 分量的入射粒子，以及 (b) 碰撞参数 $b$（初始 $x$ 值）。不需要改变初始 $y$, 但它应该足够大，使得 $\\mathrm{PE} / \\mathrm{KE} \\leq 10^{-10}$, 这意味着 $\\mathrm{KE} \\simeq E$.\n",
    "3. 好的参数是$m=0.5，v_y(0)=0.5，v_x(0)=0.0，\\Delta b=0.05，-1 \\leq b \\leq 1$.一旦你发现了快速变化的区域，你可能想降低能量并使用更精细的步长。\n",
    "4. 绘制一些显示常见和不常见行为的轨迹$[x(t), y(t)]$。特别是，绘制那些发生反向散射的轨迹，因此，绘制那些发生多次散射的轨迹。\n",
    "5. 绘制多个相空间轨迹$[x(t), \\dot{x}(t)]$和$[y(t), \\dot{y}(t)]$。这些轨迹与束缚态的轨迹有何不同？\n",
    "6. 通过确定散射粒子离开相互作用区域后的速度，即$\\mathrm{PE} / \\mathrm{KE} \\leq 10^{-10}$，确定散射角$\\theta=\\operatorname{atan2}\\left(v_x, v_y\\right)$。\n",
    "7. 确定轨迹的哪些特征会导致$d \\theta / d b$的不连续性，从而产生$\\sigma(θ)$。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.2 多粒子问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2.1 分子动力学"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "分子动力学（MD）是一种强大的模拟技术，用于研究固体、液体、非晶材料和生物分子的物理性质和化学过程。虽然我们知道量子力学是分子相互作用的正确理论，但MD方法以牛顿定律为基础，侧重于材料的整体性能。\n",
    "\n",
    "MD对牛顿定律的解决方案在概念上很简单，但当应用于大量粒子时，它就变成了“地狱般的高中物理问题”。为了不必求解描述现实样本的$10^{23}-10^{25}$个运动方程，必须进行一些近似，而是将问题限制如蛋白质模拟的约$10^6$个粒子或材料模拟的约$10^8$个粒子。如果我们取得了一些成功，那么加入更多的粒子或更多的量子力学方面的修正，则模型将会有所改进。\n",
    "\n",
    "\n",
    "在许多方面，MD模拟和后面将要介绍蒙特卡洛方法(MC)有些类似。他们都是用于处理大N个粒子的相互作用，起始处于一些组态构形，最终达到某种热平衡动态中。然而MD属于微正则系综，其系统的能量、体积和粒子数固定。而MC则相当于考虑体现与一热库相接触，这属于统计里的正则系综。\n",
    "\n",
    "由于分子系统是动态的，分子的速度和位置不断变化，因此我们需要及时跟踪每个分子的运动，以确定其对其他分子的影响，这些分子也在运动。在模拟运行足够长的时间稳定后，我们将计算动态量的时间平均值，以推断热力学性质。以氩分子系统为例，我们应用牛顿定律，假设每个分子的合力是所有其他（N-1）分子的二体力的总和：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "& m \\frac{d^2 \\mathbf{r}_i}{d t^2}=\\mathbf{F}_i\\left(\\mathbf{r}_0, \\ldots, \\mathbf{r}_{N-1}\\right) \\\\\n",
    "& m \\frac{d^2 \\mathbf{r}_i}{d t^2}=\\sum_{i<j=0}^{N-1} \\mathbf{f}_{i j}, \\quad i=0, \\ldots,(N-1) .\n",
    "\\end{aligned}\n",
    "$$\n",
    "在写这些方程式时，我们忽略了氩原子之间的力实际上是\n",
    "由构成原子的18个电子和原子核的相互作用产生。\n",
    "尽管在通常情况下,研究各类长程性质时可以忽略这种内部结构。\n",
    "但这对于多原子分子等系统很重要，这些系统在温度升高时会表现出旋转、振动和电子自由度。\n",
    "\n",
    "我们假设分子i上的力来自一个势能，而这个势能是分子-分子中心势能的总和：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "& \\mathbf{F}_i\\left(\\mathbf{r}_0, \\mathbf{r}_1, \\ldots, \\mathbf{r}_{N-1}\\right)=-\\nabla_{\\mathbf{r}_i} U\\left(\\mathbf{r}_0, \\mathbf{r}_1, \\ldots, \\mathbf{r}_{N-1}\\right), \\\\\n",
    "& U\\left(\\mathbf{r}_0, \\mathbf{r}_1, \\ldots, \\mathbf{r}_{N-1}\\right)=\\sum_{i<j} u\\left(r_{i j}\\right)=\\sum_{i=0}^{N-2} \\sum_{j=i+1}^{N-1} u\\left(r_{i j}\\right) \\\\\n",
    "& \\Rightarrow \\quad \\mathbf{f}_{i j}=-\\frac{d u\\left(r_{i j}\\right)}{d r_{i j}} \\quad\\left(\\frac{x_i-x_j}{r_{i j}} \\hat{\\mathbf{e}}_x+\\frac{y_i-y_j}{r_{i j}} \\hat{\\mathbf{e}}_y+\\frac{z_i-z_j}{r_{i j}} \\hat{\\mathbf{e}}_z\\right) .\n",
    "\\end{aligned}\n",
    "$$\n",
    "这里$r_{i j}=\\left|\\mathbf{r}_i-\\mathbf{r}_j\\right|=r_{j i}$代表分子中心的距离。求和中的限制为了防止对相互作用重复计数。因为我们假设为保守势，系统的总能量，即所有粒子的势能加动能之和，应随时间守恒。然而，在实际计算中，当分子相距较远时，我们“切断势能” （即设$u(r_{ij})=0$）。\n",
    "因为势的导数在这个截止点处产生无限大的力，能量将\n",
    "不再精确地守恒。然而，由于截止半径较大，截止仅发生在\n",
    "当力很小时。相对于近似值和舍入误差,违反能量守恒的情况应该很小\n",
    "\n",
    "为了简化计算，我们没有采用Hartree-Fock或密度泛函理论。简单设势为Lennard-Jones势\n",
    "$$\n",
    "\\begin{gathered}\n",
    "u(r)=4 \\epsilon\\left[\\left(\\frac{\\sigma}{r}\\right)^{12}-\\left(\\frac{\\sigma}{r}\\right)^6\\right] \\\\\n",
    "\\mathbf{f}(r)=-\\frac{d u}{d r} \\frac{\\mathbf{r}}{r}=\\frac{48 \\epsilon}{r^2}\\left[\\left(\\frac{\\sigma}{r}\\right)^{12}-\\frac{1}{2}\\left(\\frac{\\sigma}{r}\\right)^6\\right] \\mathbf{r} .\n",
    "\\end{gathered}\n",
    "$$\n",
    "这里参数$\\epsilon$代表了相互作用强度，而$\\sigma$决定了长度尺度。这两个参数都需要从实验测量数据中拟合出来。\n",
    "$$\n",
    "\\begin{array}{lccccc}\n",
    "\\hline \\text { 物理量 } & \\text { 质量 } & \\text { 长度 } & \\text { 能量 } & \\text { 时间 } & \\text { 温度 } \\\\\n",
    "\\text { 单位 } & m & \\sigma & \\epsilon & \\sqrt{m \\sigma^2 / \\epsilon} & \\epsilon / k_B \\\\\n",
    "\\text { 值 } & 6.7 \\times 10^{-26} \\mathrm{~kg} & 3.4 \\times 10^{-10} \\mathrm{~m} & 1.65 \\times 10^{-21} \\mathrm{~J} & 4.5 \\times 10^{-12} \\mathrm{~s} & 119 \\mathrm{~K} \\\\\n",
    "\\hline\n",
    "\\end{array}\n",
    "$$\n",
    "为了使程序更简单，避免下溢和溢出，以这些常数形成的自然单位来表示所有变量是有帮助的。则粒子间势和力的形式为\n",
    "$$\n",
    "u(r)=4\\left[\\frac{1}{r^{12}}-\\frac{1}{r^6}\\right], \\quad f(r)=\\frac{48}{r}\\left[\\frac{1}{r^{12}}-\\frac{1}{2 r^6}\\right]\n",
    "$$\n",
    "势的形势如图：\n",
    "\n",
    "![image.png](./pic/Lennard-Jones.png)\n",
    "\n",
    "该势体现为长程的吸引相互作用$\\propto 1/r^6$,短程的排斥相互作用$\\propto 1/r^{12}$\n",
    "\n",
    "与热力学变量的联系\n",
    "\n",
    "我们假设粒子的数量足够大，可以用统计力学来关联我们对热力学量的模拟结果（模拟对任何数量的粒子都有效，但使用统计需要大量粒子）。能均分定理告诉我们，对于在温度T下处于热平衡的分子，每个分子自由度的平均能量为$k_B T /2$，其中$k_B = 1.38 \\times 10^{-23} J/K$是玻尔兹曼常数。模拟给出平动动能：\n",
    "$$\n",
    "\\mathrm{KE}=\\frac{1}{2}\\left\\langle\\sum_{i=0}^{N-1} v_i^2\\right\\rangle .\n",
    "$$\n",
    "动能的时间平均值与温度的关系为\n",
    "$$\n",
    "\\langle\\mathrm{KE}\\rangle=N \\frac{3}{2} k_B T \\Rightarrow T=\\frac{2\\langle\\mathrm{KE}\\rangle}{3 k_B N} .\n",
    "$$\n",
    "由位力定理给出体系的压强为\n",
    "$$\n",
    "P V=N k_B T+\\frac{w}{3}, \\quad w=\\left\\langle\\sum_{i<j}^{N-1} \\mathbf{r}_{i j} \\cdot \\mathbf{f}_{i j}\\right\\rangle\n",
    "$$\n",
    "其中位力$w$是力的加权平均值。\n",
    "\n",
    "为了节约计算资源，我们可以采用周期性边界条件。为了启动我们的计算，需要给定初始条件。这里我们可以根据系统总能量，产生粒子的位置及速度的分布。而对于势场我们也需要引入一个截断。\n",
    "\n",
    "真实的MD模拟可能需要对$10^3-10^6$个粒子的三维运动方程进行$10^{10}$步的时间积分。尽管我们可以使用标准的rk4求解，但通过使用程序中嵌入的简单规则可以节省时间。Verlet算法使用二阶导数的中心差分近似，以同时将所有N个粒子的解推进一个时间步长h：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathbf{F}_i[\\mathbf{r}(t), t] & =\\frac{d^2 \\mathbf{r}_i}{d t^2} \\simeq \\frac{\\mathbf{r}_i(t+h)+\\mathbf{r}_i(t-h)-2 \\mathbf{r}_i(t)}{h^2} \\\\\n",
    "\\Rightarrow \\quad \\mathbf{r}_i(t+h) & \\simeq 2 \\mathbf{r}_i(t)-\\mathbf{r}_i(t-h)+h^2 \\mathbf{F}_i(t)+\\mathrm{O}\\left(h^4\\right),\n",
    "\\end{aligned}\n",
    "$$\n",
    "注意，即使原子-原子力没有明确的时间依赖性，我们在其中包含一个t依赖性，指示它在特定时间对原子位置的依赖性。因为这实际上是一个隐含的时间依赖性，所以能量是守恒的。\n",
    "\n",
    "Verlet算法的效率在于，它求解每个粒子的位置，而不需要单独求解粒子的速度。一旦我们推导了不同时间的位置，我们就可以使用$r_i$一阶导数的中心差分近似来获得速度：\n",
    "$$\n",
    "\\mathbf{v}_i(t)=\\frac{d \\mathbf{r}_i}{d t} \\simeq \\frac{\\mathbf{r}_i(t+h)-\\mathbf{r}_i(t-h)}{2 h}+\\mathrm{O}\\left(h^2\\right)\n",
    "$$\n",
    "最后要注意的是，因为Verlet算法需要来自前两个步骤的r，所以它不是自启动的，我们用前向差来启动它，\n",
    "$$\n",
    "\\mathbf{r}(t=-h) \\simeq \\mathbf{r}(0)-h \\mathbf{v}(0)+\\frac{h^2}{2} \\mathbf{F}(0)\n",
    "$$\n",
    "\n",
    "\n",
    "改进：Velocity-Verlet 算法\\\\\n",
    "稳定性更高的算法。它对导数使用向前差分近似来同时推进位置和速度\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathbf{r}_i(t+h) & \\simeq \\mathbf{r}_i(t)+h \\mathbf{v}_i(t)+\\frac{h^2}{2} \\mathbf{F}_i(t)+\\mathrm{O}\\left(h^3\\right), \\\\\n",
    "\\mathbf{v}_i(t+h) & \\simeq \\mathbf{v}_i(t)+h \\overline{\\mathbf{a}(t)}+\\mathrm{O}\\left(h^2\\right) \\\\\n",
    "& \\simeq \\mathbf{v}_i(t)+h\\left[\\frac{\\mathbf{F}_i(t+h)+\\mathbf{F}_i(t)}{2}\\right]+\\mathrm{O}\\left(h^2\\right) .\n",
    "\\end{aligned}\n",
    "$$\n",
    "尽管该算法的阶数似乎低于前一种，但在计算速度时，使用了更新的位置\n",
    "以及随后对这些速度的使用，使得这两种算法具有同阶的精度。\n",
    "\n",
    "上式将时间步长期间的平均力近似为$(F_i(t+h) + F_i(t))/2$。更新速度会有点棘手，因为我们需要时间t+h的力，这取决于t+h处的粒子位置。因此，在我们更新任何速度之前,必须更新所有粒子，将位置和力保存到t+h，同时在上式中使用的较早时间。一旦位置更新，我们就应用周期性边界条件，以确保没有损失任何粒子，然后我们计算力。\n",
    "\n",
    "练习：\n",
    "1. 可以运行并可视化一维模拟。\n",
    "\n",
    "2. 将粒子最初放置在一个简单的立方体晶格的位置。在低温下的Lennard-Jones系统平衡构形\n",
    "是面心立方(FCC)，如果模拟运行正常，那么粒子应该从SC(体立方)迁移到FCC。粒子会从SC中找到自己的演化方式。FCC晶格每个晶胞有四个粒子，因此具有晶格常数L/N的$L^3$的晶格包含$4N^3\n",
    "=32、108、256$个粒子。\n",
    "\n",
    "3. 为了节省计算时间，将初始粒子速度指定为温度麦克斯韦分布。\n",
    "\n",
    "4. 一个典型的时间步长是$∆t=10^{−14}$s，在我们的自然单位中等于0.004。你可能需要运行$10^4–10^5$步才能到达平衡态。\n",
    "\n",
    "5. 观察演化中动能KE和势能PE随t的变化。\n",
    "\n",
    "6. 计算平衡时的系统温度，并与设定的初始温度做比较。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\t PE =  (-28.515433935546852, 1048.1085427734374) \t KE =  1.1644878970481058 \t PE + KE =  [ -27.35094604 1049.27303067]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/wv/hkd30zd55sv0pgsvs7tr8rq00000gn/T/ipykernel_17784/1021398396.py:116: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  plt.plot(t_range,avT[1:]/t_range)\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.animation import FuncAnimation\n",
    "\n",
    "Natom = 25\n",
    "h = 0.031\n",
    "L = int(Natom**0.5)\n",
    "x = np.zeros(Natom)\n",
    "y = np.zeros(Natom)\n",
    "vx = np.zeros(Natom)\n",
    "vy = np.zeros(Natom)\n",
    "fx = np.zeros((Natom,2))\n",
    "fy = np.zeros((Natom,2))\n",
    "\n",
    "\n",
    "def init():\n",
    "    global x,y,vx,vy,fx,fy\n",
    "    k = 0\n",
    "    for i in range(0,L):\n",
    "        for j in range(0,L):\n",
    "            x[k] = i\n",
    "            y[k] = j\n",
    "            vx[k] = np.random.randn()\n",
    "            vy[k] = np.random.randn()\n",
    "            k=k+1\n",
    "\n",
    "    sigma=np.sqrt(2) # (kT/m)**(1/2)\n",
    "    mu = 0\n",
    "    vx = sigma*np.random.randn(Natom) + mu\n",
    "    vy = sigma*np.random.randn(Natom) + mu\n",
    "\n",
    "def sign(a,b):\n",
    "    if (b >= 0 ):\n",
    "          return abs(a)\n",
    "    else:\n",
    "        return - abs(a)\n",
    "\n",
    "def force(t):\n",
    "    global x,y,vx,vy,fx,fy\n",
    "    r2cut = 9.\n",
    "    PE = 0.\n",
    "    w = 0.\n",
    "    ftol_x = np.zeros(Natom)\n",
    "    ftol_y = np.zeros(Natom)\n",
    "    fx[:,t] = 0.\n",
    "    fy[:,t] = 0.\n",
    "    for i in range(Natom):\n",
    "        for j in range(i+1,Natom):\n",
    "                delta_x = x[i] - x[j]\n",
    "                delta_y = y[i] - y[j]\n",
    "                if abs(delta_x) > L/2:\n",
    "                    delta_x = delta_x - sign(L,delta_x)\n",
    "                if abs(delta_y) > L/2:\n",
    "                    delta_y = delta_y - sign(L,delta_y)\n",
    "                r2 = delta_x**2 + delta_y**2\n",
    "                if (r2 < r2cut):\n",
    "                    if (r2 == 0):\n",
    "                         r2 = 0.0001\n",
    "                    r = np.sqrt(delta_x**2 + delta_y**2)\n",
    "                    f = 48/r * (1/r**12 - 1/(2 * r**6))\n",
    "                    fx[i,t] += f * (delta_x / r)\n",
    "                    fy[i,t] += f * (delta_y / r)\n",
    "                    fx[j,t] -= f * (delta_x / r)\n",
    "                    fy[j,t] -= f * (delta_y / r)\n",
    "                    PE = PE + 4. * (1/r**6)*(1/r**6 - 1)\n",
    "                    w = w + f*r\n",
    "    return PE,w\n",
    "\n",
    "def calc_KE():\n",
    "    return np.sum(vx**2 + vy**2) / 2\n",
    "\n",
    "def update():\n",
    "    global x,y,vx,vy,fx,fy\n",
    "    for i in range(0,Natom):\n",
    "        force(0)\n",
    "        x[i] = x[i] + h*vx[i] + h**2 / 2 * fx[i,0] \n",
    "        y[i] = y[i] + h*vy[i] + h**2 / 2 * fy[i,0] \n",
    "        if x[i] <=0:\n",
    "            x[i] = x[i] + L\n",
    "        if x[i] >=L:\n",
    "            x[i] = x[i] - L\n",
    "        if y[i] <=0:\n",
    "            y[i] = y[i] + L\n",
    "        if y[i] >=L:\n",
    "            y[i] = y[i] - L\n",
    "    PE,w = force(1)\n",
    "\n",
    "    vx = vx + h/2 * (fx[:,0] + fx[:,1])\n",
    "    vy = vy + h/2 * (fy[:,0] + fy[:,1])\n",
    "    KE = calc_KE()\n",
    "    P = KE + w\n",
    "    T = KE / (Natom)\n",
    "    return P,T,PE,KE \n",
    "\n",
    "\n",
    "\n",
    "def time_evolution():\n",
    "    global x,y,vx,vy,fx,fy\n",
    "    avT = [0.0]\n",
    "    avP = [0.0]\n",
    "    avKE = [0.0]\n",
    "    avPE = [0.0]\n",
    "\n",
    "    init()\n",
    "    KE = np.average((vx**2 + vy**2)/2)\n",
    "    PE = force(0)\n",
    "    print(\"\\t PE = \",PE,\"\\t KE = \",KE,\"\\t PE + KE = \",(PE+KE))\n",
    "    t_range = np.arange(1000)\n",
    "    for it in t_range:\n",
    "        P,T,PE,KE = update()\n",
    "        avT.append(avT[-1] + T)\n",
    "        avP.append(avP[-1] + P)\n",
    "        avKE.append(avKE[-1] + KE)\n",
    "        avPE.append(avPE[-1] + PE)\n",
    "    \n",
    "    plt.plot(t_range,avT[1:]/t_range)\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "time_evolution()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2.2 蒙特卡洛模拟"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 如何产生随机数，以及如何判断这些随机数的质量\n",
    "2. 如何用随机数去模拟物理过程\n",
    "3. 如何使用随机数模拟热力学过程及量子系统的涨落\n",
    "\n",
    "\n",
    "关于随机数的一些概念：如果一组序列中的数之间没有关联，那么它们是随机的。如果一组序列中的所有数出现的可能性相等，那么它们是均一的(uniform)。随机变量；概率分布函数；均匀分布；正态分布。\n",
    "\n",
    "计算机无法产生真正的随机序列。通过算法产生出来数称为伪随机数。这些数之间隐含者关联性。一种方式是通过一些自然的过程，如放射性衰变，来获取随机数。\n",
    "\n",
    "伪随机数产生：线性同余法或幂余法，可以生成\n",
    "$0 \\le r_i \\le M-1$区间的随机数序列。\n",
    "$$\n",
    "r_{i+1} \\stackrel{\\text { def }}{=}\\left(a r_i+c\\right) \\bmod M\n",
    "$$\n",
    "例如设$c=1, a=4, M=9$，并提供初始值$r_1 = 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "c,a,M=1,4,9\n",
    "r1 = 3\n",
    "r = r1\n",
    "for i in range(10):\n",
    "    r = (a*r+c)%M \n",
    "    print(r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以看出序列的长度为$M = 9$。设$0 \\le r \\le 1$。获得其他区间的随机数\n",
    "$$\n",
    "x_i=A+(B-A) r_i, \\quad 0 \\leq r_i \\leq 1, \\Rightarrow A \\leq x_i \\leq B\n",
    "$$\n",
    "\n",
    "随机性和均匀性测试\n",
    "\n",
    "因为计算机的随机数是根据一定的规则生成的，序列中的数字必然相互关联。这可能会影响假设随机事件的模拟。测试随机数生成器以获得其均匀性和随机性的数值测量是明智的。事实上，一些测试很简单，可以养成同时运行它们和模拟的习惯。在接下来的例子中，我们测试随机性或均匀性。\n",
    "\n",
    "1. 最简单的，输出这些数字看一下，不会有明显的模式。\n",
    "2. 通过图形画出$r_i$随i的变化。同样不应该有明显的模式。\n",
    "3. 计算分布的k阶矩测试其均匀性\n",
    "$$\n",
    "\\left\\langle x^k\\right\\rangle=\\frac{1}{N} \\sum_{i=1}^N x_i^k\n",
    "$$\n",
    "如果分布是均匀的，那么则有\n",
    "$$\n",
    "\\frac{1}{N} \\sum_{i=1}^N x_i^k \\simeq \\int_0^1 d x x^k P(x) \\simeq \\frac{1}{k+1}+O\\left(\\frac{1}{\\sqrt{N}}\\right)\n",
    "$$\n",
    "如果满足该关系，则分布是均匀的。而如果偏离的变化满足$1/\\sqrt{N}$,则同样可以认为分布是随机的。\n",
    "4. 另一个简单的测试是通过计算小部分的乘积之和来确定随机序列中的近邻相关性\n",
    "$$\n",
    "C(k)=\\frac{1}{N} \\sum_{i=1}^N x_i x_{i+k}, \\quad(k=1,2, \\ldots)\n",
    "$$\n",
    "如果分布满足均匀性，则有\n",
    "$$\n",
    "\\frac{1}{N} \\sum_{i=1}^N x_i x_{i+k} \\simeq \\int_0^1 d x \\int_0^1 d y x y P(x, y)=\\frac{1}{4}\n",
    "$$\n",
    "\n",
    "测试：采用第三步的方式，测试你的随机数产生子\n",
    "$$\n",
    "\\sqrt{N}\\left|\\frac{1}{N} \\sum_{i=1}^N x_i^k-\\frac{1}{k+1}\\right|\n",
    "$$\n",
    "它应该接近于1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "蒙特卡洛的应用--随机游走\n",
    "\n",
    "考虑一个在教室前方释放的香水分子。它随机地与空气中的其他分子碰撞，并最终到达你的鼻子，即使你躲在最后一排。问题是确定一个香水分子在行进距离R时平均会发生多少次碰撞。已知一个分子在碰撞之间的平均（均方根）距离为$r_{rms}$。\n",
    "\n",
    "我们考虑最简单的二维扩展随机游走。在我们的随机游走模拟中，一个步行者按照顺序的步骤进行，每一步的方向独立于前一步的方向。对于我们的模型，我们从原点开始，在xy平面内进行N步的长度\n",
    "$$\n",
    "\\left(\\Delta x_1, \\Delta y_1\\right),\\left(\\Delta x_2, \\Delta y_2\\right),\\left(\\Delta x_3, \\Delta y_3\\right), \\ldots,\\left(\\Delta x_N, \\Delta y_N\\right)\n",
    "$$\n",
    "尽管每一步可能朝着不同的方向，但沿着每个笛卡尔坐标轴的距离只是代数上的增加。因此，N步后距离起点的径向距离R为\n",
    "$$\n",
    "\\begin{aligned}\n",
    "R^2= & \\left(\\Delta x_1+\\Delta x_2+\\cdots+\\Delta x_N\\right)^2+\\left(\\Delta y_1+\\Delta y_2+\\cdots+\\Delta y_N\\right)^2 \\\\\n",
    "= & \\Delta x_1^2+\\Delta x_2^2+\\cdots+\\Delta x_N^2+2 \\Delta x_1 \\Delta x_2+2 \\Delta x_1 \\Delta x_3+2 \\Delta x_2 \\Delta x_1+\\cdots \\\\\n",
    "& +(x \\rightarrow y)\n",
    "\\end{aligned}\n",
    "$$\n",
    "如果行走是随机的，粒子在每一步都有同样的可能向任何方向移动。如果我们取大量随机步的平均值，则上式中的所有交叉项都将消失，我们得到\n",
    "$$\n",
    "\\begin{aligned}\n",
    "R_{\\mathrm{rms}}^2 & \\simeq\\left\\langle\\Delta x_1^2+\\Delta x_2^2+\\cdots+\\Delta x_N^2+\\Delta y_1^2+\\Delta y_2^2+\\cdots+\\Delta y_N^2\\right\\rangle \\\\\n",
    "& =\\left\\langle\\Delta x_1^2+\\Delta y_1^2\\right\\rangle+\\left\\langle\\Delta x_2^2+\\Delta y_2^2\\right\\rangle+\\cdots \\\\\n",
    "& =N\\left\\langle r^2\\right\\rangle=N r_{\\mathrm{rms}}^2, \\\\\n",
    "& \\Rightarrow R_{\\mathrm{rms}} \\simeq \\sqrt{N} r_{\\mathrm{rms}},\n",
    "\\end{aligned}\n",
    "$$\n",
    "其中$r_{rms}=\\sqrt{<r^2>}$为步长均方根。\n",
    "\n",
    "总之，如果步行是随机的，那么我们预期在经过大量步数后，与原点的平均矢量距离将消失：\n",
    "$$\n",
    "\\langle\\vec{R}\\rangle=\\langle x\\rangle \\vec{i}+\\langle y\\rangle \\vec{j} \\simeq 0\n",
    "$$\n",
    "\n",
    "然而距原点的平均标量距离为$\\sqrt{N}r_{rms}$，其中每一步的平均长度为$r_{rms}$ 。换句话说，向量端点将在所有象限中均匀分布，因此位移向量平均为零，但该向量的长度不为零。对于较大的N值，$\\sqrt{N}r_{rms}\\ll Nr_{rms}$但不会消失。根据经验，实际模拟与该理论一致，但很少完全一致，一致程度取决于如何取平均值以及如何在每一步中引入随机性的细节。\n",
    "\n",
    "练习：\n",
    "\n",
    "1. 为了增加随机性，在[-1,1]的范围内独立选择$\\Delta x^{\\prime}$和$\\Delta y^{\\prime}$的随机值。然后对其进行归一化，使每一步的长度为1$$\\Delta x=\\frac{1}{L} \\Delta x^{\\prime}, \\quad \\Delta y=\\frac{1}{L} \\Delta y^{\\prime}, \\quad L=\\sqrt{\\Delta x^{\\prime 2}+\\Delta y^{\\prime 2}} .$$\n",
    "2. 使用绘图程序绘制几个独立的二维随机游走图，每个游走图有1000步。使用模拟得出的证据，评论这些游走图是否与你对随机游走的预期相符。\n",
    "3. 如果在一次试验中走了 $N$ 步，那么进行总计 $K \\simeq \\sqrt{N}$ 次试验。每次试验应有 $N$ 步，并从不同的种子开始\n",
    "4.  计算每个试验的均方距离 $R^2$，然后取所有 $K$ 个试验的 $R^2$ 的平均值：$$\\left\\langle R^2(N)\\right\\rangle=\\frac{1}{K} \\sum_{k=1}^K R_{(k)}^2(N) .$$\n",
    "5. 重复上述过程，并进行三维游走分析。\n",
    "6. 通过检查以下假设的有效性来检查推导理论结果的假设的有效性$$\\frac{\\left\\langle\\Delta x_i \\Delta x_{j \\neq i}\\right\\rangle}{R^2} \\simeq \\frac{\\left\\langle\\Delta x_i \\Delta y_j\\right\\rangle}{R^2} \\simeq 0 .$$检查单次（长）运行和试验平均值。\n",
    "7. 将均方根距离$R_{\\mathrm{rms}}=\\sqrt{\\left\\langle R^2(N)\\right\\rangle}$绘制为$\\sqrt{N}$的函数。 $N$的值应从小数开始，其中$R \\simeq \\sqrt{N}$预计不准确，最后是一个相当大的值，其中平均值预计有两到三个精度位。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "蒙特卡洛积分\n",
    "如何测量不规则池塘的面积？\n",
    "\n",
    "很难相信蒙特卡洛技术可以用于计算积分。毕竟，我们不想在数值上赌博！虽然其他方法更适合单变量和双变量积分，但当积分的维数变大时，蒙特卡洛技术是最好的！对于我们的池塘问题，我们将使用抽样技术\n",
    "\n",
    "1. 移来一个完全封闭池塘的箱子，并清除箱子内地面上的所有鹅卵石。\n",
    "2. 以自然单位（如英尺）测量边的长度。这将告诉封闭框$A_｛ext｛box｝｝$的面积。\n",
    "3. 抓起一堆鹅卵石，数一数它们的数量，然后朝空中随意抛出。\n",
    "4. 计算池塘中溅起的水花数量 $N_{\\text {pond }}$ 和你箱子内地面上鹅卵石的数量 $N_{\\text {box }}$.假设你均匀随机地抛掷鹅卵石，落入池塘的鹅卵石数量应与池塘面积$A_{\\text {pond }}$成正比。你根据简单的比率确定该面积$$\\frac{N_{\\text {pond }}}{N_{\\text {pond }}+N_{\\text {box }}}=\\frac{A_{\\text {pond }}}{A_{\\text {box }}} \\Rightarrow A_{\\text {pond }}=\\frac{N_{\\text {pond }}}{N_{\\text {pond }}+N_{\\text {box }}} A_{\\text {box }} \\text {. }$$\n",
    "\n",
    "![imag.png](./pic/sec3-2-montecarlo.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "蒙特卡洛积分技术的标准是基于平均值定理：$$I=\\int_a^b dx f(x)=(b-a)\\langle f\\rangle$$如果你把积分看作面积，这个定理就显而易见了：在a和b之间的某个函数f(x)的积分的值等于区间(b-a)的长度乘以该函数在该区间上的平均值$\\langle f\\rangle$。积分算法使用蒙特卡罗技术来计算平均值。使用N个均匀随机数的序列a≤x_i≤b，我们想通过在这些点上采样函数f(x)来确定样本均值：$$\\langle f\\rangle \\simeq \\frac{1}{N} \\sum_{i=1}^N f\\left(x_i\\right)$$这给了我们一个非常简单的积分规则：$$\\int_a^b d x f(x) \\simeq(b-a) \\frac{1}{N} \\sum_{i=1}^N f\\left(x_i\\right)=(b-a)\\langle f\\rangle .$$\n",
    "\n",
    "方程看起来很像我们用于积分的标准算法，其中“点”$x_i$ 是随机选择的，权重 $w_i=(b-a) / N$ 均匀分布。这并不是计算积分的最佳方法；但你会承认它很简单。如果我们让 $f(x)$ 的样本数量接近无穷大 $N \\rightarrow \\infty$，或者我们保持样本数量有限并取无穷多次运行的平均值，统计定律保证将接近正确答案。在f(x)的N个样本之后，积分I的值的不确定性由标准偏差$\\sigma_I$来衡量。如果$\\sigma_f$是采样中积分项f的标准偏差，那么对于正态分布，我们有$$\\sigma_I \\simeq \\frac{1}{\\sqrt{N}} \\sigma_f$$因此，对于较大的$N$，积分值的误差随着$1 / \\sqrt{N}$而减小。\n",
    "\n",
    "\n",
    "假设我们想要计算一个12个电子的小原子（如镁）的一些性质。为此，我们需要将原子波函数在12个电子的每个坐标上进行积分。这相当于一个3×12=36维的积分。如果每维积分使用64个点，则需要大约$64^36$次对被积分的求值。如果计算机速度很快，每秒可以计算被积分的100万次，那么这将花费大约$10^59$秒，这比宇宙的年龄（约$10^17$秒）长得多。\n",
    "\n",
    "现在的问题是要找到一种方法来执行多维积分，这样仍然可以活着去获得答案。具体来说，计算10维积分$$I=\\int_0^1 dx_1 \\int_0^1 dx_2 \\cdots \\int_0^1 dx_{10}\\left(x_1+x_2+\\cdots+x_{10}\\right)^2 .$$将数值答案与解析答案 $\\frac{155}{6}$ 进行比较。\n",
    "\n",
    "当我们进行多维积分时，蒙特卡洛技术的误差是统计性的，误差随 $1 / \\sqrt{N}$ 而减小。即使 $N$ 个点分布在 $D$ 个维度上，这也是有效的。相比之下，当我们使用这些相同的 $N$ 个点，通过辛普森等规则将 $D$ 维积分作为 $D$ 个 1 积分进行计算时，我们每次积分使用 $N / D$ 个点。对于固定的 $N$，这意味着每次积分使用的点数随着维度 $D$ 的增加而减少，因此每次积分的误差随着维度增加而增大。此外，总误差将大约是每个积分误差的 $N$ 倍。如果我们把这些放在一起，并考虑一个特定的积分规则，我们会发现，当 $D \\simeq 3-4$ 时，蒙特卡洛积分的误差与常规方案的误差相似。对于更大的 $D$ 值，蒙特卡洛方法总是更准确！\n",
    "\n",
    "\n",
    "练习：\n",
    "\n",
    "使用内置随机数生成器计算上面的10-D蒙特卡罗积分。\n",
    "1. 进行 16 次试验，取平均值作为答案。 \n",
    "2. 尝试样本大小$N=2,4,8, \\ldots, 8192$。\n",
    "3. 绘制误差与$1 / \\sqrt{N}$的关系图，看看是否出现线性行为。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果被积函数变化比较剧烈，例如物理中常见的高斯型，则之前的方式需要非常多的点才能到达一个满意的精度。\n",
    "\n",
    "方差减少法\n",
    "\n",
    "第一种方法是方差减少或减法技术，其中我们设计了一个更平坦的函数，在该函数上应用蒙特卡洛技术。假设我们在[a, b]上构造了一个具有以下性质的函数$g(x)$：$$|f(x)-g(x)| ≤ \\epsilon, \\quad \\int_a^b d x g(x)=J .$$我们现在计算$f(x)-g(x)$的积分，并将结果加到$J$中，以获得所需的积分$$\\int_a^b d x f(x)=\\int_a^b d x[f(x)-g(x)]+J。$$如果我们足够聪明，找到一个简单的$g(x)$，使得$f(x)-g(x)$的方差小于$f(x)$的方差，并且我们可以进行解析积分，我们可以在更短的时间内获得更精确的答案。\n",
    "\n",
    "第二种改进蒙特卡罗积分的方法称为重点采样，因为它允许我们在最重要的区域对被积函数进行采样。它源于以如下形式表示积分$$I=\\int_a^b dx f(x)=\\int_a^b dx w(x) \\frac{f(x)}{w(x)}。$$如果我们现在使用$w(x)$作为随机数的加权函数或概率分布，则积分可以近似为$$I=\\left\\langle\\frac{f}{w}\\right\\rangle \\simeq \\frac{1}{N} \\sum_{i=1}^N \\frac{f\\left(x_i\\right)}{w\\left(x_i\\right)}$$改进的是，明智地选择加权函数$w(x) \\propto f(x)$使得$f(x) / w(x)$更恒定，从而更容易积分。\n",
    "\n",
    "\n",
    "冯·诺伊曼推导出了一个简单而巧妙的方法，可以生成具有概率分布$w(x)$的随机点。这种方法与用于猜测池塘面积的拒绝或采样方法基本相同，只不过现在池塘被加权函数$w(x)$取代，湖周围的任意盒子被任意常数$W_0$取代。想象一下w(x)与x的图\n",
    "![imag.png](./pic/sec3-2-importance_sampling.png)。\n",
    "通过在图上放置线W=W_0，唯一的条件是W_0≥w(x)。接下来，我们在这个图上“扔石头”，只计算那些落入w(x)池塘的石头。也就是说，我们在x和y=W中生成均匀分布，最大y值等于盒子的宽度W_0：$$\\left(x_i, W_i\\right)=\\left(r_{2 i-1}, W_0 r_{2 i}\\right) .$$然后，我们拒绝所有不属于池塘的$x_i$：$$如果 W_i<w\\left(x_i\\right) ，则接受，如果 W_i>w\\left(x_i\\right) ，则拒绝。$$如此接受的$x_i$值将具有权重$w(x)$。最大接受率发生在$w(x)$较大的情况下，在这种情况下为中值$x$。\n",
    "\n",
    "如何用冯·诺伊曼法产生正态分布？\n",
    "\n",
    "反变换/变量转换法：让我们考虑将原始积分转换为如下形式的变量转换$$I=\\int_a^b dx f(x)=\\int_0^1 d W \\frac{f[x(W)]}{w[x(W)]}。$$我们的目标是进行这种转换，使W范围内所有部分的贡献相等；也就是说，我们希望对W使用统一的随机数序列。为了确定新变量，我们首先使用u(r)，即[0,1]上的均匀分布，$$u(r)= \\begin{cases}1, & \\text { for } 0 \\leq r \\leq 1 \\\\ 0, & \\text { otherwise }\\end{cases}$$我们想找到一个映射 $r \\leftrightarrow x$ 或概率函数 $w(x)$，其中概率是守恒的：$$w(x) d x=u(r) d r，\\quad \\Rightarrow \\quad w(x)=\\left|\\frac{d r}{d x}\\right| u(r) .$$这意味着，即使 $x$ 和 $r$ 之间存在某种（可能）复杂的映射关系，但 $x$ 也是随机的，其概率等于 $r$ 位于 $r \\rightarrow r+d r$ 的概率。为了找到 $x$ 和 $r$ 之间的映射（最棘手的部分），我们将变量改为由积分定义的 $$W(x)=\\int_{-\\infty}^x d x^{\\prime} w\\left(x^{\\prime}\\right)$$我们认识到$W(x)$是概率密度$u(r)$在某一点$x$之前的（不完全）积分。它是另一种分布函数，是找到小于值$x$的随机数的积分概率。因此，函数$W(x)$被称为累积分布函数，也可以被看作是u(r)与r的图上r=x左侧的面积。根据定义，$W(x)\n",
    "具有以下特性$$\\begin{aligned}W(-\\infty) & =0；\\quad W(\\infty)=1\\frac{d W(x)}{d x} & =w(x), \\quad d W(x)=w(x) d x=u(r) d r .\\end{aligned}$$因此，$W_i=\\left\\{r_i\\right\\}$是一个均匀的随机数序列，我们只需要进行求逆，即可得到以概率$w(x)$分布的$x$值。这种技术的关键在于能够求逆以获得$x=W^{-1}(r)$.让我们来看一些解析示例，以了解这些步骤（在现实情况下，数值求逆是可能且频繁的）。\n",
    "\n",
    "高斯（正态）分布：我们想生成正态分布的点：$$w\\left(x^{\\prime}\\right)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} e^{-\\left(x^{\\prime}-\\bar{x}\\right)^2 / 2 \\sigma^2} .$$这本身是相当困难的，但通过在角度上生成均匀分布，然后使用三角关系将其转换为高斯分布，使问题变得更容易。但在这样做之前，我们通过实现用均值 $\\bar{x}$ 和标准差 $\\sigma$ 通过缩放和翻译一个更简单的 $w(x)$ 来获得 (6.73)，从而使事情变得简单：$$w(x)=\\frac{1}{\\sqrt{2 \\pi}} e^{-x^2 / 2}, x^{\\prime}=\\sigma x+\\bar{x} .$$我们首先将两个不同分布的概率守恒陈述（6.63）推广到二维[Pres 94]：$$\\begin{aligned}p(x, y) d x d y & = & u\\left(r_1, r_2\\right) d r_1 d r_2 \\\\\\Rightarrow \\quad p(x, y) & = & u\\left(r_1, r_2\\right)\\left|\\frac{\\partial\\left(r_1, r_2\\right)}{\\partial(x, y)}\\right| .\\end{aligned}$$比值是雅可比行列式：$$J=\\left|\\frac{\\partial\\left(r_1, r_2\\right)}{\\partial(x, y)}\\right| \\stackrel{\\text{def}}{=} \\frac{\\partial r_1}{\\partial x} \\frac{\\partial r_2}{\\partial y}-\\frac{\\partial r_2}{\\partial x} \\frac{\\partial r_1}{\\partial y}。$$为了专注于高斯分布，我们将$2 \\pi r$视为从均匀随机分布$r$中获得的角度，而将$x$和$y$视为具有高斯分布的笛卡尔坐标。两者之间的关系如下$$x=\\sqrt{-2 \\ln r_1} \\cos 2 \\pi r_2, \\quad y=\\sqrt{-2 \\ln r_1} \\sin 2 \\pi r_2 .$$此映射的反演产生高斯分布$$r_1=e^{-\\left(x^2+y^2\\right) / 2}, \\quad r_2=\\frac{1}{2 \\pi} \\tan ^{-1} \\frac{y}{x}, \\quad J=-\\frac{e^{-\\left(x^2+y^2\\right) / 2}}{2 \\pi} .$$使用该公式，$r_1$和$r_2$由均匀随机分布产生，然后$x$和$y$是中心在$x=0$的高斯随机分布。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2.3 量子蒙特卡洛模拟"
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "问题：\n",
    "\n",
    "铁磁体包含有限大小的畴，畴中所有原子的自旋指向同一方向。当外部磁场施加到这些材料上时，不同的畴变得排列整齐，材料变得“磁化”。然而，随着温度的升高，总磁性降低，在居里温度下，系统经历相变，超过相变后所有磁化消失。问题是如何解释铁磁体的热行为。\n",
    "\n",
    "模型（Ising 模型）：\n",
    "\n",
    "作为我们的模型，我们认为N个磁偶极子固定在线性链的链结上!（这是处理二维和三维晶格的直接推广。）\n",
    "![sec3-15.1.png](attachment:sec3-15.1.png)\n",
    "\n",
    "因为粒子是固定的，它们的位置和动量不是动态变量，我们只需要担心它们的自旋。我们假设位于i位的粒子具有自旋$s_i$，它要么向上要么向下：\n",
    "$$\n",
    "s_i \\equiv s_{z, i}= \\pm \\frac{1}{2}\n",
    "$$\n",
    "N个粒子的每种组态由量子态矢量描述\n",
    "$$\n",
    "\\left|\\alpha_j\\right\\rangle=\\left|s_1, s_2, \\ldots, s_N\\right\\rangle=\\left\\{ \\pm \\frac{1}{2}, \\pm \\frac{1}{2}, \\ldots\\right\\}, \\quad j=1, \\ldots, 2^N\n",
    "$$\n",
    "因为每个粒子的自旋可以取两个值中的任何一个，所以系统中N个粒子有2N个不同的可能状态。由于固定的粒子不能互换，我们不需要关心波函数的对称性。\n",
    "\n",
    "系统的能量来自自旋之间的相互作用以及与外部磁场B的相互作用。我们从量子力学中知道，电子的自旋和磁矩是相互成比例的，因此磁偶极子-偶极子相互作用等价于自旋-自旋相互作用。我们假设每个偶极子通过势能与外部磁场及其最近邻相互作用：\n",
    "$$\n",
    "V_i=-J \\mathbf{s}_i \\cdot \\mathbf{s}_{i+1}-g \\mu_b \\mathbf{s}_i \\cdot \\mathbf{B}\n",
    "$$\n",
    "这里常数J被称为交换能量，是自旋-自旋相互作用强度的量度。常数g是旋磁比，即粒子的角动量和磁矩之间的比例常数。常数$\\mu_b=e \\hbar /\\left(2 m_e c\\right)$是玻尔磁子，是磁矩的基本量度。\n",
    "\n",
    "即使对于少量的粒子，$2^N$个可能的自旋组态也会变得非常大（$2^{20}>10^6$），并且计算机检查所有这些组态的成本很高。具有约$10^{23}$个粒子的现实样本超出了想象。因此，即使对于中等大小的N，通常也假设使用统计方法。为了准确起见，N必须有多大？这是希望能在模拟中探索的事情之一。\n",
    "\n",
    "该系统在状态$\\alpha_k$下的能量是粒子自旋势V的和的期望值：\n",
    "$$\n",
    "E_{\\alpha_k}=\\left\\langle\\alpha_k\\left|\\sum_i V_i\\right| \\alpha_k\\right\\rangle=-J \\sum_{i=1}^{N-1} s_i s_{i+1}-B \\mu_b \\sum_{i=1}^N s_i .\n",
    "$$\n",
    "当我们关闭外部磁场并因此消除空间中的优先方向时，伊辛模型中会出现明显的悖论。这意味着即使最低能量状态的所有自旋都排列整齐，平均磁化强度也会消失。这个悖论的答案是，B = 0的系统是不稳定的。即使所有自旋都排列整齐，也没有什么可以阻止它们自发反转。事实上，天然磁性材料具有多个有限区域，所有自旋都排列整齐，但不同区域指向不同方向。区域改变方向的不稳定性被称为“畴变”。为了简单起见，我们假设B = 0，这意味着只有自旋相互作用。但是，要知道这意味着空间中没有优先方向，因此计算可观测量时必须小心。例如，在计算磁化强度时，可能需要取总自旋的绝对值，即计算$\\left\\langle\\left|\\sum_i s_i\\right|\\right\\rangle$而不是$\\left\\langle\\sum_i s_i\\right\\rangle$。\n",
    "\n",
    "自旋的平衡排列在很大程度上取决于交换能 J 的符号。如果 J > 0，最低能量状态将倾向于使相邻的自旋排列。如果温度足够低，基态将是一个所有自旋都排列整齐的铁磁体。如果 J < 0，最低能量状态将倾向于使相邻的自旋相反。如果温度足够低，基态将是一个自旋交替排列的反铁磁体。\n",
    "\n",
    "简单的1维Ising模型有其局限性。虽然该模型在描述热平衡系统方面是准确的，但在描述趋向热平衡的方面是不准确的（非平衡热力学是一个困难的课题，其理论是不完整的）。其次，作为我们算法的一部分，我们假设一次只翻转一个自旋，而真实的磁性材料倾向于一次翻转许多自旋。其他局限性很容易改进，例如，添加长程相互作用、中心运动、更高多重性的自旋态以及二维和三维。\n",
    "\n",
    "\n",
    "理论（统计物理）：\n",
    "\n",
    "\n",
    "统计力学从系统粒子之间的基本相互作用开始，构建宏观热力学性质，如比热。基本假设是系统所有符合约束的组态都是可能的。在一些模拟中，如前面“用分子动力学模拟物质”中的分子动力学模拟，问题的设置是系统的能量是固定的。这种系统的状态由所谓的微正则系综描述。相比之下，对于本章研究的热力学模拟，温度、体积和粒子数保持不变，因此我们拥有所谓的正则系综。\n",
    "\n",
    "当我们说一个物体处于温度T时，我们意味着该物体的原子在温度T下处于热力学平衡状态，使得每个原子的平均能量与T成正比。虽然这可能是一个平衡状态，但它是一个动态状态，在该状态下，物体的能量随着它与环境交换能量而波动（毕竟这是热力学）。事实上，我们在模拟中最具启发性的方面之一是它在平衡状态下发生的持续和随机能量交换，这还可以进行可视化。\n",
    "\n",
    "正则系综中状态$\\alpha_j$的能量$E_{\\alpha_j}$不是常数，而是以玻尔兹曼分布给出的概率$P(\\alpha_j)$分布：\n",
    "$$\n",
    "\\mathcal{P}\\left(E_{\\alpha_j}, T\\right)=\\frac{e^{-E_{\\alpha_j} / k_B T}}{Z(T)}, \\quad Z(T)=\\sum_{\\alpha_j} e^{-E_{\\alpha_j} / k_B T}\n",
    "$$\n",
    "其中k是玻尔兹曼常数，T是温度，Z(T)是配分函数，是各态的加权和。注意，式中的和是系统各态或各构型的和。另一种表述，如后面介绍的Wang-Landau算法，是系统各态能量的和，并包括态密度因子$g(E_i)$，以考虑具有相同能量的简并态。虽然目前的状态和能量之和是表达问题的更简单的方法（少了一个函数），但我们将看到能量之和在数值上更有效。事实上，在这里，我们甚至忽略了配分函数Z(T)，因为在处理概率之比时它会被抵消。\n",
    "\n",
    "解析解：\n",
    "\n",
    "\n",
    "对于非常大量的粒子，一维伊辛模型的热力学性质可以通过解析方法求解，并确定\n",
    "$$\n",
    "\\begin{gathered}\n",
    "U=\\langle E\\rangle \\\\\n",
    "\\frac{U}{J}=-N \\tanh \\frac{J}{k_B T}=-N \\frac{e^{J / k_B T}-e^{-J / k_B T}}{e^{J / k_B T}+e^{-J / k_B T}}= \\begin{cases}N, & k_B T \\rightarrow 0, \\\\\n",
    "0, & k_B T \\rightarrow \\infty\\end{cases}\n",
    "\\end{gathered}\n",
    "$$\n",
    "每个粒子的比热和磁化率的解析结果为\n",
    "$$\n",
    "\\begin{aligned}\n",
    "C\\left(k_B T\\right) & =\\frac{1}{N} \\frac{d U}{d T}=\\frac{\\left(J / k_B T\\right)^2}{\\cosh ^2\\left(J / k_B T\\right)} \\\\\n",
    "M\\left(k_B T\\right) & =\\frac{N e^{J / k_B T} \\sinh \\left(B / k_B T\\right)}{\\sqrt{e^{2 J / k_B T} \\sinh ^2\\left(B / k_B T\\right)+e^{-2 J / k_B T}}}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "二维伊辛模型有一个解析解，但很难推导出来。虽然内能和热容量都是用椭圆积分表示的，但每个粒子的自发磁化强度却有相当简单的形式\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathcal{M}(T) & = \\begin{cases}0, & T>T_c \\\\\n",
    "\\frac{\\left(1+z^2\\right)^{1 / 4}\\left(1-6 z^2+z^4\\right)^{1 / 8}}{\\sqrt{1-z^2}}, & T<T_c\\end{cases} \\\\\n",
    "k T_c & \\simeq 2.269185 J, \\quad z=e^{-2 J / k_B T},\n",
    "\\end{aligned}\n",
    "$$\n",
    "其中温度以居里温度$T_c$为单位测量。\n",
    "\n",
    "算法(Metropolis 算法):\n",
    "\n",
    "\n",
    "在尝试设计模拟热平衡的算法时，重要的是要理解玻尔兹曼分布并不要求系统保持在最低能量状态，而是说系统处于较高能量状态的可能性低于较低能量状态。当然，随着$T\\to 0$，只有最低能量状态会被填充。对于有限温度，我们预期能量在平衡值附近波动约$k_B T$。\n",
    "\n",
    "在他们对中子穿过物质的模拟中，Metropolis、Rosenbluth、Teller和Teller发明了一种算法来改进蒙特卡罗平均值计算。这种Metropolis算法现在是计算物理学的基石。它产生的组态序列（马尔可夫链）准确地模拟了热平衡过程中发生的波动。该算法随机改变单个自旋，使得平均而言，组态发生的概率遵循玻尔兹曼分布。\n",
    "\n",
    "Metropolis算法是上节中讨论的方差缩减技术和冯·诺依曼拒绝技术的组合。在那里，我们展示了如何通过主要在积分项较大的地方采样随机点来使蒙特卡洛积分更有效，以及如何生成具有任意概率分布的随机点。现在我们希望随机翻转自旋，使系统能够在有限步骤内达到任何能量（遍历采样），使能量分布由玻尔兹曼分布描述，但使系统能够足够快地达到平衡，以便在合理的时间内进行计算。\n",
    "\n",
    "Metropolis算法是通过多个步骤实现的。我们从一个固定的温度和初始自旋组态开始，应用该算法直到达到热平衡（平衡）。算法的持续应用产生了关于平衡的统计波动，从中我们推断出热力学量，如磁化强度M（T）。然后改变温度，重复整个过程，以推断出热力学量的T依赖性。推断出的温度依赖性的准确性为该算法的有效性提供了令人信服的证据。由于N个粒子的可能$2^N$个可能组态是一个非常大的数字，所需的计算机时间可能很长。通常，少量迭代$\\simeq 10N$足以达到平衡。\n",
    "\n",
    "Metropolis算法的明确步骤如下。\n",
    "1. 从任意自旋配置开始，即$\\alpha_k=\\left\\{s_1, s_2, \\ldots, s_N\\right\\}$.\n",
    "2. 通过以下方式生成试验配置 $\\alpha_{k+1}$\n",
    "   a. 随机选取一个粒子 $i$，\n",
    "   b. 翻转它的旋转。\n",
    "3. 计算试验配置的能量$E_{\\alpha_{\\mathrm{tr}}}$。\n",
    "4. 如果$E_{\\alpha_{\\mathrm{tr}}} \\leq E_{\\alpha_k}$,则通过设置$\\alpha_{k+1}=\\alpha_{\\mathrm{tr}}$来接受试验。\n",
    "5. 如果$E_{\\alpha_{\\mathrm{tr}}}>E_{\\alpha_k}$,则以相对概率$\\mathcal{R}=\\exp \\left(-\\Delta E / k_B T\\right)$接受：\n",
    "   a. 选择一个均匀随机数$0 \\leq r_i \\leq 1$。\n",
    "   b. 设置$\\alpha_{k+1}= \\begin{cases}\\alpha_{\\mathrm{tr}}, & \\text { 如果 } \\mathcal{R} \\geq r_j \\text { (接受)， } \\\\ \\alpha_k, & \\text { 如果 } \\mathcal{R}<r_j \\text { (拒绝)。 }\\end{cases}$\n",
    "\n",
    "该算法的核心是它以概率产生随机自旋组态$\\alpha_j$\n",
    "$$\n",
    "\\mathcal{P}\\left(E_{\\alpha_j}, T\\right) \\propto e^{-E_{\\alpha_j} / k_B T} .\n",
    "$$\n",
    "该技术是冯·诺伊曼拒绝法的变体，其中随机试验组态根据玻尔兹曼因子的值被接受或拒绝。明确地说，能量$E_t$的试验组态与能量$E_i$的初始组态的概率之比为\n",
    "$$\n",
    "\\mathcal{R}=\\frac{\\mathcal{P}_{\\mathrm{tr}}}{\\mathcal{P}_i}=e^{-\\Delta E / k_B T}, \\quad \\Delta E=E_{\\alpha_{\\mathrm{tr}}}-E_{\\alpha_i}\n",
    "$$\n",
    "如果试验组态的能量较低$\\Delta E \\le 0$，相对概率将大于1，我们将接受试验组态作为新的初始组态，无需进一步处理。然而，如果试验组态的能量较高$\\Delta E > 0$，我们不会立即拒绝它，而是以相对概率$\\mathcal{R}=\\exp \\left(-\\Delta E / k_B T\\right)<1$接受它。为了以概率接受配置，我们选择一个介于0和1之间的均匀随机数，如果概率大于这个数字，我们接受试验配置；如果概率小于选定的随机数，我们拒绝它。当试验组态被拒绝时，下一个组态与前一个组态相同。\n",
    "\n",
    "如何开始？一种可能是从自旋的随机值开始（“热”开始）。另一种可能是“冷”开始，其中从所有自旋平行（J>0）或反平行（J<0）开始。一般来说，在计算平衡热力学量之前，人们会通过让计算运行一段时间（$\\simeq 10N$次重排）来消除起始配置的重要性。你应该得到热、冷或任意启动的类似结果，并通过取它们的平均值来消除一些统计波动。\n",
    "\n",
    "算法实现：\n",
    "\n",
    "1. 编写一个实现 Metropolis 算法的程序，即从当前配置 $\\alpha_k$ 生成新的配置 $\\alpha_{k+1}$。\n",
    "2. 将程序中的关键数据结构设为数组 $\\mathbf{s}[\\mathrm{N}]$，其中包含自旋 $s_i$ 的值。为了调试，打印出 + 和 -，以给出每个格点处的自旋，并检查不同试验数的模式。\n",
    "3. 交换能量的值 $J$ 确定了能量的标度。将其固定在 $J=1$。\n",
    "4. 热能$k_B T$的单位是$J$，是自变量。使用$k_B T=1$进行调试。\n",
    "5. 在链上使用周期性边界条件以最小化端部效应。这意味着链是一个圆圈，第一个和最后一个旋转彼此相邻。\n",
    "6. 尝试使用 $N \\simeq 20$ 进行调试，并在生产运行中使用更大的值。\n",
    "7. 使用打印输出检查系统是否平衡\n",
    "   a. 完全有序的初始配置（冷启动）。\n",
    "   b. 随机初始配置（热启动）。\n",
    "\n",
    "\n",
    "练习:\n",
    "\n",
    "\n",
    "1. 观察N个原子在与热浴接触时达到热平衡。在高温下，或对于少数原子，应该看到较大的波动，而在较低的温度下，应该看到较小的波动。\n",
    "2. 寻找大量自旋自发翻转的不稳定证据。对于较大的$k_B T$值，这种情况更有可能发生。\n",
    "3. 注意在热平衡状态下，系统仍然具有很强的动态性，自旋一直在翻转。正是这种能量交换决定了热力学性质。\n",
    "4. 你可能会发现，在较小的 $k_B T$（例如，对于 $N=200$，$k_B T \\simeq 0.1$）下，模拟达到平衡的速度很慢。较高的 $k_B T$ 值达到平衡的速度更快，但波动更大。\n",
    "5.观察畴的形成及其对总能量的影响。无论畴内的自旋方向如何，原子-原子相互作用都是吸引的，因此在排列时对系统的能量有负贡献。然而，畴之间的$\\uparrow \\downarrow$或$\\downarrow \\uparrow$相互作用贡献了正能量。因此，在温度较低时，畴更大更少，能量应更负。\n",
    "6. 绘制平均畴的尺寸与温度的图表。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "Nspin = 30\n",
    "Bfield = 1.\n",
    "mu = 0.33\n",
    "J = 0.2\n",
    "k_B = 1\n",
    "T = 100\n",
    "state = np.zeros((Nspin))\n",
    "Spin = np.zeros((Nspin))\n",
    "test = state\n",
    "np.random.seed()\n",
    "\n",
    "def energy(S):\n",
    "    FirstTerm = 0.0\n",
    "    SecondTerm = 0.0\n",
    "    for i in range(0,Nspin-2):\n",
    "        FirstTerm += S[i]*S[i+1]\n",
    "    FirstTerm *=-J\n",
    "    for i in range(0,Nspin-1):\n",
    "        SecondTerm +=S[i]\n",
    "    SecondTerm *= - Bfield*mu\n",
    "    return  (FirstTerm + SecondTerm)\n",
    "\n",
    "ES = energy(state)\n",
    "\n",
    "\n",
    "for i in range(0,Nspin):\n",
    "    state[i] = -1\n",
    "\n",
    "Es = energy(state)\n",
    "\n",
    "for j in range(1,500):\n",
    "    test = state\n",
    "    r = int(Nspin*np.random.random())\n",
    "    test[r] *= -1\n",
    "    ET = energy(test)\n",
    "    p = np.exp(Es - ET)/ (k_B * T)\n",
    "    if p >= np.random.random():\n",
    "        state = test\n",
    "        Es = ET\n",
    "    "
   ]
  },
  {
   "attachments": {
    "sec3-15.6.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wang-Landau 采样\n",
    "\n",
    "\n",
    "在上一单元中，我们使用玻尔兹曼分布来模拟伊辛模型的热性质。在那里，我们描述了温度为T的系统在能量为$E_\\alpha$时自旋态$\\alpha$的概率，并求出了各种组态的总和。一个等效公式描述了系统在温度T下具有显式能量E的概率：\n",
    "$$\n",
    "\\mathcal{P}\\left(E_i, T\\right)=g\\left(E_i\\right) \\frac{e^{-E_i / k_B T}}{Z(T)}, \\quad Z(T)=\\sum_{E_i} g\\left(E_i\\right) e^{-E_i / k_B T}\n",
    "$$\n",
    "这里$k_B$是玻尔兹曼常数，T是温度，$g(E_i)$是能量$E_i$的状态数（i=1，…，M），Z(T)是配分函数，并且总和仍然是系统所有M个状态的总和，但由于$g(E_i)$考虑了它们的简并性，因此相同能量的状态只出现一次。由于我们再次将该理论应用于具有离散自旋状态的伊辛模型，因此能量也只取离散值。如果物理系统具有连续变化的能量，则$E\\to E+dE$区间内的状态数将由g(E)dE给出，g(E)被称为状态密度。为了方便起见，即使在处理离散系统时，我们也称$g(E_i)$为状态密度，尽管“简并因子”一词可能更精确。\n",
    "\n",
    "尽管Metropolis算法已经提供了50多年的优质服务，但最近的文献表明，Wang-Landau采样（WLS）的使用越来越多。由于WLS确定了状态密度和相关的配分函数，它不是Metropolis算法及其热涨落模拟的直接替代品。然而，我们将看到WLS为Ising模型提供了等效模拟。而在某些情况下，Metropolis算法可以使用，但WLS不能，谐振子基态函数的计算就是一个例子。\n",
    "\n",
    "WLS的优点是它比Metropolis算法需要更短的模拟时间，并且可以直接确定$g(E_i)$。由于这些原因，已经证明它对于系统长时间陷入亚稳态的一级相变特别有用，以及在自旋系统、流体、液晶、聚合物和蛋白质等不同领域。模拟所需的时间在模拟大型系统时变得至关重要。即使像8×8这样小的自旋晶格也有$2^{64}\\simeq 1.84\\times 10^{19}$个组态，并且访问所有组态的成本太高。\n",
    "\n",
    "在上一单元中，我们在使用Metropolis算法时忽略了配分函数。现在我们关注配分函数Z(T)和态密度函数g(E)。因为g(E)是能量的函数，而不是温度的函数，一旦推导出来，Z(T)和所有热力学量都可以从中计算出来，而不需要对每个温度重复模拟。例如，内能和熵直接计算为\n",
    "$$\n",
    "\\begin{aligned}\n",
    "U(T) & \\stackrel{\\text { def }}{=}\\langle E\\rangle=\\frac{\\sum_{E_i} E_i g\\left(E_i\\right) e^{-E_i / k_B T}}{\\sum_{E_i} g\\left(E_i\\right) e^{-E_i / k_B T}}, \\tag{20} \\\\\n",
    "S & =k_B \\ln g\\left(E_i\\right) .\n",
    "\\end{aligned}\n",
    "$$\n",
    "态密度$g(E_i)$通过在能量空间中进行随机游走来决定。我们随机翻转一个自旋，记录新组态的能量，然后通过翻转更多自旋来改变能量，从而继续行走。每个能量$E_i$达到次数的表$H(E_i)$被称为能量直方图。如果行走持续很长时间，直方图$H(E_i)$将收敛到态密度$g(E_i)$。然而，即使对于小型系统，也需要$10^{19}-10^{30}$步，这种直接方法效率低得离谱，因为行走很少会脱离最可能的能量。\n",
    "\n",
    "王-朗道算法背后的第一个聪明想法是通过增加进入不太可能组态的可能性来探索更多的能量空间。这是通过增加不太可能组态的接受度，同时减少更可能组态的接受度来实现的。换句话说，我们希望接受密度$g(E_i)$较小的更多状态，拒绝密度$g(E_i)$较大的更多状态。为了实现这一技巧，我们以与状态密度成反比的概率接受新的能量$E_i$，\n",
    "$$\n",
    "\\mathcal{P}\\left(E_i\\right)= \\frac{1}{g(E_i)}\n",
    "$$\n",
    "然后通过随机游走建立起获得的态的直方图。\n",
    "\n",
    "聪明想法1的问题在于$g(E_i)$是未知的。WLS的聪明想法2是在构建随机游走的同时确定未知的$g(E_i)$。这是通过将$g(E_i)$乘以$f g(E_i)$来提高$g(E_i)$的值来实现的，其中$f>1$是一个经验因子。当这起作用时，得到的直方图$H(E_i)$会变得“更平坦”，因为使较小的$g(E_i)$值变大，使其更有可能达到具有较小$g(E_i)$值的状态。随着直方图变得更平坦，我们不断减小乘法因子f，直到它接近1。此时，我们有一个平坦的直方图和确定的$g(E_i)$。\n",
    "\n",
    "此时，你可能会问，为什么平直的直方图意味着我们已经确定了$g(E_i)$？平直意味着所有能量都被平等地访问，与通常在没有$1/g(E_i)$加权因子的情况下获得的尖峰直方图形成鲜明对比。因此，如果通过包含这个加权因子，我们产生了平直的直方图，那么我们就完美地抵消了$g(E_i)$中的实际尖峰，这意味着我们已经得到了正确的$g(E_i)$。\n",
    "\n",
    "\n",
    "WLS算法实现\n",
    "\n",
    "WLS中的步骤与Metropolis算法中的步骤类似，但现在使用态密度函数$g\\left(E_i\\right)$而不是玻尔兹曼因子：\n",
    "1. 从任意自旋组态$\\alpha_k=\\left\\{s_1, s_2, \\ldots, s_N\\right\\}$开始，状态密度g（E_i）=1，i=1,…,M，其中M=2^N是系统的状态数。\n",
    "2. 通过以下方式生成试验组态 $\\alpha_{k+1}$\n",
    "    1. 随机选取一个粒子 $i$，\n",
    "    2. 翻转$i$的旋转。\n",
    "3. 计算试验构型的能量$E_{\\alpha_{\\mathrm{tr}}}$。\n",
    "4. 如果$g\\left(E_{\\alpha_{\\mathrm{tr}}}\\right) \\leq g\\left(E_{\\alpha_k}\\right)$,则接受试验，即设置$\\alpha_{k+1}=\\alpha_{\\mathrm{tr}}$.\n",
    "5. 如果$g\\left(E_{\\alpha_{\\text {tr }}}\\right)>g\\left(E_{\\alpha_k}\\right)$,则以概率$ \\mathcal{P}=g\\left(E_{\\alpha_k}\\right) / \\left(g\\left(E_{\\alpha_{\\text {tr }}}\\right)\\right.$接受试验：\n",
    "   1. 选择一个均匀的随机数$0 \\leq r_i \\leq 1$。\n",
    "   2. 设置$\\alpha_{k+1}= \\begin{cases}\\alpha_{\\mathrm{tr}}, & \\text { 如果 } \\mathcal{P} \\geq r_j \\text { (接受)， } \\\\ \\alpha_k, & \\text { 如果 } \\mathcal{P}<r_j \\text { (拒绝)。 }\\end{cases}$这个接受规则可以简洁地表示为\n",
    "$$\n",
    "\\mathcal{P}\\left(E_{\\alpha_k} \\rightarrow E_{\\alpha_{\\mathrm{tr}}}\\right)=\\min \\left[1, \\frac{g\\left(E_{\\alpha_k}\\right)}{g\\left(E_{\\alpha_{\\mathrm{tr}}}\\right)}\\right], \\tag{23}\n",
    "$$\n",
    "其显然总是接受低密度(不太可能)状态。\n",
    "6. 当我们有一个新的状态时，我们通过乘法因子$f$修改当前的态密度$g\\left(E_i\\right)$：\n",
    "$$\n",
    "g\\left(E_{\\alpha_{k+1}}\\right) \\rightarrow fg\\left(E_{\\alpha_{k+1}}\\right) \\tag{24}\n",
    "$$\n",
    "并将1添加到直方图中与新能量对应的容器中：\n",
    "$$\n",
    "H\\left(E_{\\alpha_{k+1}}\\right) \\rightarrow H\\left(E_{\\alpha_{k+1}}\\right)+1\n",
    "$$\n",
    "7. 乘数$f$的值是经验性的。我们以欧拉数$f=e=$ 2.71828开始，它似乎在大量小步（小$f$）和通过能量空间的一组过快跳跃（大$f$）之间取得了很好的平衡。因为熵$S=k_B \\ln g\\left(E_i\\right) \\rightarrow k_B\\left[\\ln g\\left(E_i\\right)+\\ln f\\right]$, (24)对应于熵均匀增加$k_B$。\n",
    "8. 即使$f$的值为合理值，(24)中的重复乘法也会导致$g$的幅值呈指数增长。这可能会导致浮点溢出和信息一致丢失[最终，由于函数已归一化，因此$g\\left(E_i\\right)$的幅值并不重要]。通过使用函数值的对数来避免这些溢出，在这种情况下，状态密度的更新(24)变为\n",
    "$$\n",
    "\\ln g\\left(E_i\\right) \\rightarrow \\ln g\\left(E_i\\right)+\\ln f .\n",
    "$$\n",
    "9. 存储$g\\left(E_i\\right)$的难点在于我们需要$g\\left(E_i\\right)$值的比率来计算(23)中的概率。通过使用恒等式$x=\\exp (\\ln x)$将比率表示为\n",
    "$$\n",
    "\\frac{g\\left(E_{\\alpha_k}\\right)}{g\\left(E_{\\alpha_{\\mathrm{tr}}}\\right)}=\\exp \\left[\\ln \\frac{g\\left(E_{\\alpha_k}\\right)}{g\\left(E_{\\alpha_{\\mathrm{tr}}}\\right)}\\right]=\\exp \\left[\\ln g\\left(E_{\\alpha_k}\\right)\\right]-\\exp \\left[\\ln g\\left(E_{\\alpha_{\\mathrm{tr}}}\\right)\\right] \\text {。 }\n",
    "$$\n",
    "反过来，$g\\left(E_k\\right)=f \\times g\\left(E_k\\right)$被修改为$ln g\\left(E_k\\right) \\rightarrow \\ln g\\left(E_k\\right)+\\ln f$。\n",
    "10.  $E_i$中的随机游走会一直持续，直到获得访问能量值的扁平直方图。直方图的扁平度会定期（每10,000次迭代）进行测试，一旦直方图足够扁平，游走就会终止。然后，$f$的值会减小，以便下一次游走能更好地近似$g\\left(E_i\\right)$.通过比较$H\\left(E_i\\right)$的方差与其平均值来测量扁平度。虽然像我们这样的小问题可以实现90%-95%的扁平度，但我们只需要80%：\n",
    "如果$\\frac{H_{\\max }-H_{\\min }}{H_{\\max }+H_{\\min }}<0.2$,停止，让$f \\rightarrow \\sqrt{f}(\\ln f \\rightarrow \\ln f / 2)$.\n",
    "11.  然后保留生成的$g\\left(E_i\\right)$并将直方图值$h\\left(E_i\\right)$重置为零。\n",
    "12.  终止走步并开始新的走步，直到没有获得对态密度的显著校正。这是通过要求乘法因子$f \\simeq 1$在一定容差范围内来测量的；例如，$f \\leq 1+10^{-8}$.如果算法成功，直方图应在第10步设定的范围内保持平坦。\n",
    "13. 模拟的最后一步是对推导出的态密度$g\\left(E_i\\right)$进行归一化。对于具有N个向上或向下自旋的伊辛模型，根据态总数的知识，归一化条件如下：\n",
    "$$\n",
    "\\sum_{E_i} g\\left(E_i\\right)=2^N \\Rightarrow g^{(\\text {norm })}\\left(E_i\\right)=\\frac{2^N}{\\sum_{E_i} g\\left(E_i\\right)} g\\left(E_i\\right) .\n",
    "$$\n",
    "因为上式中的总和受那些能量值的影响最大，其中$g\\left(E_i\\right)$很大，因此对于低能量密度（对总和贡献很小）可能不精确。因此，更精确的归一化，要求只有两个基态，能量为$E=-2 N$（一个所有自旋都向上，一个所有自旋都向下）：\n",
    "$$\n",
    "∑_{E_i=-2 N} g\\left(E_i\\right)=2\n",
    "$$\n",
    "在任何一种情况下，用一种条件对$g\\left(E_i\\right)$进行归一化，然后用另一种条件进行检验，这是一种很好的做法。\n",
    "\n",
    "\n",
    "WLS 在Ising模型中的应用\n",
    "\n",
    "我们假设一个Ising模型，在位于$L \\times L$格点中的最近邻之间存在自旋-自旋相互作用。\n",
    "![sec3-15.6.png](attachment:sec3-15.6.png)\n",
    "\n",
    "为了使符号简单，我们设定$J=1$，因此\n",
    "$$\n",
    "E=-∑_{i \\leftrightarrow j}^N \\sigma_i \\sigma_j，\n",
    "$$\n",
    "其中 $\\leftrightarrow$ 表示最近邻。不是每次翻转自旋时都重新计算能量，而是只计算能量的差值。例如，对于一维数组中的八个自旋，\n",
    "$$\n",
    "-E_k=\\sigma_0 \\sigma_1+\\sigma_1 \\sigma_2+\\sigma_2 \\sigma_3+\\sigma_3 \\sigma_4+\\sigma_4 \\sigma_5+\\sigma_5 \\sigma_6+\\sigma_6 \\sigma_7+\\sigma_7 \\sigma_0\n",
    "$$\n",
    "其中$0-7$相互作用是由于我们假设了周期性边界条件而产生的。如果自旋5被翻转，则新的能量为\n",
    "$$\n",
    "-E_{k+1}=\\sigma_0 \\sigma_1+\\sigma_1 \\sigma_2+\\sigma_2 \\sigma_3+\\sigma_3 \\sigma_4-\\sigma_4 \\sigma_5-\\sigma_5 \\sigma_6+\\sigma_6 \\sigma_7+\\sigma_7 \\sigma_0\n",
    "$$\n",
    "能量差为\n",
    "$$\n",
    "\\Delta E=E_{k+1}-E_k=2\\left(\\sigma_4+\\sigma_6\\right) \\sigma_5。\n",
    "$$\n",
    "这比计算并减去两个能量更便宜。\n",
    "当我们使用图中的8×8格子进行二维计算时，自旋$\\sigma_{i,j}$翻转时的能量变化为\n",
    "$$\n",
    "\\Delta E=2 \\sigma_{i,j}\\left(\\sigma_{i+1, j}+\\sigma_{i-1, j}+\\sigma_{i, j+1}+\\sigma_{i, j-1}\\right)\n",
    "$$\n",
    "可以假设这些值是$-8,-4,0,4$和8。对于一个具有N个自旋的二维系统，有两种最低能量状态，即$-2 N$，它们对应于所有自旋指向同一方向（向上或向下）。最大能量是$+2 N$,它对应于相邻位置上自旋方向的交替。晶格上的每次自旋翻转都会使这些极限之间的能量变化四个单位，因此能量的值是\n",
    "$$\n",
    "E_i=-2 N, -2 N+4, -2 N+8, \\ldots, 2 N-8, 2 N-4, 2 N.\n",
    "$$\n",
    "这些能量可以通过简单的映射存储在均匀的一维阵列中\n",
    "$$\n",
    "E^{\\prime}=(E+2 N) / 4 \\quad \\Rightarrow \\quad E^{\\prime}=0,1,2, \\ldots, N\n",
    "$$\n",
    "首先实现王-兰道采样计算态密度和内能$U(T)$的代码。我们再用它来获得熵$S(T)$和能量直方图$H\\left(E_i\\right)$.其他热力学函数可以通过将(20)中的$E$替换为适当的变量来获得。计算这些变量时可能遇到的问题是，即使比率不会溢出，(20)中的总和也可能变得足够大而引起溢出。你可以通过分解一个共同的大因子来解决这个问题；例如\n",
    "$$\n",
    "\\sum_{E_i} X\\left(E_i\\right) g\\left(E_i\\right) e^{-E_i / k_B T}=e^\\lambda \\sum_{E_i} X\\left(E_i\\right) e^{\\ln g\\left(E_i\\right)-E_i / k_B T-\\lambda},\n",
    "$$\n",
    "其中，$\\lambda$是每个温度下ln g(E_i)-E_i / k_B T的最大值。在变量的计算中，实际上不需要包含因子$e^\\lambda$，因为它同时出现在分子和分母中，因此可以抵消。"
   ]
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
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"
    },
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"
    },
    "sec3-15.9.png": {
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iYsN3UUTnLxkdSFAiQASIABEgAq4gIJshvzbt2sHbxwd6nQ4NGzVyBUtZtKlVadhmNQq8UaUPDkacwxt/jxGVXnEpNrYvIc2hElWnkDBEgAgQASIA2RhUlStXhodSCU+tFr6+vtS1BSTAvVU7un2PKY2H48KDq3j7n/GITIouYK0FLx72wIIVF2mVX8FJUg1EgAgQASJgTwKyMag4FJWHBwoXKWJPPm5dl5/WB+ULheJeUhT23D2FZr8PwpmoC27NhJSXDoE/31+P++fuSUdgkpQIEAFJE5CVQdWjV0907tJZ0h0iRuEbF68teKueL9cWZX1DsDF8l8u8VUX1StQrKpupf2LsbtnIdHnfRVzacVE2+pAiRIAIiJuAwsaSuEXMvXTh4eHw9PRE8eLFsy2UEpeMX19aBkNhg5Cn5sC6KN2ibLb56UFmAjfib6P06pfQ1r8sNnSZC5VCVjZ5ZmXpSrIEkmONWNhhPgKfKYme3/aVrB4kOBEgAtIhIKuf+j5sUnpifAIsJgtUOQQr8gvxR+eZPaTTSyKS9J+bB1HTEIB5LUaTMSWifiFRMhO4eeC6cCPmuuvn/WWWjK6IABGQKwFZGFRLfvwRX8/+Gl4GgxA2gXuqOnTsiJmzZ0GloiiQ9nh5ebDPVRe34K1nBqBraCvwfQFdkS7FWXE6yoRuIZ6uaJ7alAiBi9vCYLVYWVw6C5IiE6ELEF9MNYmgJDGJABHIJQHJj9ccPXIEY78Yg2effRb/7NmNzVv/QpUqVfD7+vU4efJkLjFQtpwIcGOq/5aPsfj8H4hKinGZMcVljDFacS6awibk1F/0DLh95raAwZSUgluHU71VxIUIEAEi4EgCkjeoLl68CKvVivoN6gucuEdq5arf8N3C71GzZk1HsnObuo9FnEWSJQW/tZ+KAJ2f2+hNikqTQGJEAuIiYqFUKWGz2nB+y3lpKkJSEwEiICkCkp+U/uDBA3Tr3AWFfAth3Pjx8PP3hzcb+vMvXDjLjuCT0rd+vonmUGVJJ/NN7placGo5PqnzBixs7QIP+unqZLTYkMQ2SPbzlPxvAVejlG37SVGJuHP0Fk6tO46QhqVRpEpxlKgVKFt9STEiQATEQUDy30p8IvpzLzyPE8dPYMgbb+KD94Zj3vz54qArYSnShvk2hO9DBAvoKQZjiuPUqhRkTEn4vXKG6Dp/PUq3LgefEoVQskEpMqacAZ3aIAJEAJKflD5t6lTMmzMXzz3/PG2IbMcX+npc6hyUVe2nI9BQ1I41F6wq5pyCiQ3j6JhhRYkIEAEiQASIgFgISN5D9fu69QLLtmxz5LS0detWtG3dBgcPHky7RX9zSYB7pt7dNVGIkL61+/cI8RbXUMnx+2Z8c9KYS20oGxEgAkSACBAB5xCQvEEVEhIikNq6dZvw12KxYMLY8bgQFoYjhw87h6JMWkkb5vv33hnEmOIozpRM+pXUIAJEgAgQAccTkLxBNX3mDLRp2xbr161D+7bt0Kp5C9y8eQNdu3XD668NcjxBGbUQZ0qAt1qHX9tPQymvEqLUzEetQBkfii0mys4pgFCRkZGYP2deAWqgokSACBAB1xKQ/ByqEiVKYOGiH4To6AcO/gelQoFy5cojoEjWq/xci1ucrXPP1Ed7p2Niw3ewvvMccQr5UKryvirwg5J8CPy0dCmmTZ2GQmyByVvDhshHMdKECBABtyIgeYMqrbf4VjONWHBPHkahUKFCabez/JsYnYBjSw8Jz0o1CIF/xSJZ5nOHm2nDfLsfXMc4+Wzr6A5dJxsdZ301A7H8/y0zqCgRASJABKRKQPJDfo+Dnz/36cMG3PgqVNJXONRetIVJRd9S2NF2Isw2M6KTYx9HKqrrW4kW7L1rEpVMJAwRIAJEgAgQAVkZVGFsIvqCXMSg8vTSCnFqeKwa75Lu/au4qC4A81t8gaaB9fDJvplo/fsg/HVtj2j/Z9xOsGHvbbNo5SPBiAARIAJEwD0JyMqg2vbXX0Iv3r171z17s4Baz2n+KeoUroC5J1fAxob/kq0pBayRihOB7AksXrQYw997HwmJiUKm6Oho4Zrfu3n9RvYF6QkRIAJEQIQEZDOHirPdvHkzNBoN9u7Zg17/+58IcYtbJO6t+r7leMSlJOBIxGm88c9ETKw/BO1DmopG8LKF2HCtRla/A0TD1tmC7Nm9C9u3bU9vNj4+HmtWrxauO3XujJLBQenP8nOSEp+CfTN3QWNI3TKpRI1AVH+udn6qojJEgAgQgacSkI1BxTdIvnnzFlJSUrB502YyqJ7a9dln8NYYUIoF9KzLvFVddo7HllZj0Sq4UfYFnPjEV6MAPyhJn0D7Dh3gzSaib964CUajEQa2B2fb9u0ExUoGlSywgmq9BlV6VYNfaIBQl4enbD7uCsyGKiACRMD+BGTzCXPixAlhpRBHxD1UlApGoIjOHwtajkW/m/+iRWADzDr2Iyr4lUGnkGYFq5hKE4GHBPr26wd+1Nn5j2BQ+bONzWfNnm03PgrmyNT66qEvbLBbnVQRESACRCA7ArIZO9Hr9Xht0OuoXKUyhn/wQXb60v08EmhZsqFQIiwmHN13TsDH+6bnsQb7Zj8Wacbc07T1jH2pUm1EgAgQASJQUAKy8VBVqFABAwYORHh4uGBYFRQMlX9EQMGCpc5jKwE7XdmBJEsy4s2J2HfrCNqVavIok5POLFYgwcR2SKZEBIhAjgSuXbuGYUMeBUr1MnihfMUKeIvdK168eI5l6SERIAJ5JyAbgyrvqlOJvBLoUrqVUOSXsA14af83GBTaBBMavAN/rW9eq8p3frVKAR+aQ5VvflTQfQgkstWTJ46fwNx58+HBYu/xoMcb1q/HSy+8iC1bU1dEuw8N0pQIOJ4AGVSOZyy7FgZW6MpW2nlh8pHFMDKPVWRSNAJ0fk7Rs7q/CvygRASIQO4ItOvQDmq1WsjMd5No1rgJEuLiYfD2yl0FlIsIEIFcESCDKleYKNPjBDqHtkSHkOZIsSaj2dpXUa9wJYxn3ipnGVaPy0PXRIAIZE2Ar6C0WCyIi4vD8mW/oHKlSmRMZY2K7hKBAhFwS4PKZrPCamKTcVhSeCjA5whRyjsBFVtGpVPpMLbuG/jkwDyMPzQfs5qOyntFeSgRnWzFvSQbKtIGyXmgJs2s3Ajg4VDSvCvS1ML1Uj9TpWq6EEqlElOnu3ZhSbowdEIEZEbALQ2qmOtsLsE7qQEEa/Svg9DmZWTWrc5Vp1NoCzZZPQXzTq1gf5MEI8tRElyOtWL7TRM+rqlzVBNUr4sIWNkPnYxpypdf4o8Nf2Dfv/sz3qbzPBI4G3YeGg8NYuNjsW/PPnw4/H0UKVwYzVu2yGNNlJ0IEIGcCMgmbEJOSj7+zK+UH7rP7yMcZEw9Tifv1wvP/IpX93yFYK9iSDRRSIO8E3TvEnx3A57i2LweSvYn4OHhAb4hvJ+fHzp37Ywq1aph957d9m+IaiQCbk7ALT1Ubt7ndlN/x/X90Gv0aBvcBAvZJPU+5TrZre7sKgo0qNA8MLundF+KBPQsQjpPsWwV2jvD3kZFtrS/U5cu6apcuHABO//+G8WKFkWbdu3AY85Ryh+BUydO4cL583j51VfyV0EOpW7fvo0bN26gXr16Qi4+Z+skC7jcoH5DwaDLrmhSUpKwd2jG53wahk6X2Qt96cIlJKcYUaVq6hDmnTt3cPPGTdSpWydjUTonAi4jQAaVy9BLt2EjG94bvX8mvr60HSMqdMSERu8jhG1V44xUQq9ACT29ts5g7aw2qtesgUuXLgnN/c6W9fOk8khdlXaPbXTepWMnBLAo6nfYeUhICNasXyd4W4SM9M9TCdSuUVPIYzabwSwX9B3QH127dn1qubxmUKlUGPTqa/hi7Bj07NUL773zDrRaHZ5t3DjHqhrUrQce4iFjKsS2JDp8/FjGW4h5EIPnWKzBNWvXoGy5cniu/wDWTk8yqDJRogtXEpDVN5OHUgUf9h+RkuMInI26iEr+ZYUAn8ubDEfvsh0c1xjV7BYEJoyfgFvM03D48GEIX/pM6ytXLqNQoULC9Ycff4w333oTx48fR89u3TF3zhx8+tlnbsGmIEqWLVsWu/ftTa9Cq9UiICAAfGJ6Vun69euY+80cFrvqOPoxo6v/gAHw9PTMKmuW94oyD+KkyV/i09GjcPvWLYSdD8PmLVuyzJvx5onTpzJeZnvOPVGvvf4aPhr+AarXrIlixYph6NtvZ5ufHhABZxNQ2FhydqOubC8lLhlbP9+EzjN7uFIMybUdkxyH8QfnYuGlv7G782xUZ2ESXJHOx1hwlG0/079s7j/oXSEntZl3Anxj84tseM+XGVLFAwPx5eTJWPzDIhw/dTJ9mK9dm7YwsCG/tb+nerJyauWfSdtQrW9NBJQrnFM2esYI3Lx+Ax3at8e169eQyMIs+LP5Vs82ehZrmTcwr4l7ptavWy+UrVmrlrBSMzqaxapjxlxWad3atemGdNpzDYub1a3Hk5/RfOVnt85dwI2/v3fuRECRR317l3kwuZHFk8VkESbh83ljlIiAswhk/VPFWa1n087qVauxY9v2bJ7SbWcTMNvM2H5jPzbdOIhFTd53mTHF9Y5n285ci8u8GszZPKg9xxDgk9P5/JjAoKB0L0qJEiXSjSneanBwsBBPKTcS2Kw2PGAreqMuRQlHwj2a9J4dt69mfIWrzJiKjY+Hhf3EvhcRgbNnz7DjbHZFsrzPDZkLYRfg7e2N/XtTV2dyY2rcmLFZ5uc3t2zeLKzm5Cs6046//tqaZf4HrC4+9Mti3eDkyROZ8jw/YCCLs7VcuBcefgV8lSglIuBMAqIb8ntj0GDhP9jY8eOcyYHayoJAdHIsxh6YCxX78PqqyUi0LFnfqdvMZCES3XIzArfY0JHJZBJiUXFn+rEjR1CtevVcUTAnmRG2+Sy0PqnezOJVSqBSz2q5Kutuma6HX0dKcoqg9kuvTsbypZ8jKcmI2zdvoXLlyrnGMW3aVKGvVvy6En1792GhGZqjGNs38D4z0L6cNBkBhQMwaPDgTPXN//bbTNc5Xbz//nC0bt0aTZs2wUcffoS/tm9Ln0/HvWprVv+GFq1a5FQFPSMCDiMgOg/VxbAwhylLFeeeAP/y6rv5A/x18xAaB6auonHmnn3ZSVqzsAfera7N7jHdlxkBPqdqwvjxbI7VDcHLERkVhd59++RKS7VBjXqDGqLF6LbCQcZU9tj69u+HEiVSN0xWblyAzn5F4c8WAtRv2CD7Qo89+e+///DT0qWYMXOG4Gl85bVXhVWbfCg3MjISH3z0IW4xA23vnkfzuh6rIsfLpUuWCCsUv/jiC3Tt3h18OPFDNp8qLalYeIgxY8dj1MiRabfoLxFwKgHReaicqj019gQB7pWafmwRPqzxKsY2eAvlCoWgqC7ruQ9PFHbCDbY3suAxc0JT1ISLCbA9DFCCeTd27fwHSxb/CB5PacjQoQ5ZoeZiVV3efO8+vYXJ6Js2bsQxZgDN9g2GYuIImNR6vLErHiNr6VHaO+ff38VZX234YwPKsBV4PL337vto2bIlTKy+GmwlJ494X5et6AtjP5obN8l55V9WQOrVr4/Wbdqkb5szY9ZMNrwYJsyX4nG2eKr6TFXBmFu+YkVWVdA9IuBQAqKZlH7u3DlMHDeOrfQ5IiyhLcf+U/JfTEOGDQPf0NNeiSalZ0/ySMRpvPb3OBjZnKklLb5A/WK5G1rJvkb7PzGyCR4JbB5VgDbnD3f7t0w1upIAn4TMV5HlZdUZTUrPe4/xmFB3WDypoN0HofD3g0fXdjjHFoJMOJqIGgEeeP8ZHdhuXXlK3DvVq3sPzP7ma8ybOw8jRoxE2fJl81RHbjIP7Ncfv6xcIRhY3bt1ZUPDz+DLKVNyU5TyEAG7EBDNtxIPArd79570eCQXL14UrnnwNkqOJRBtjMXBeyeFWFKtS9bGrm7fi9KY4hROR1nww7lkxwKh2kVHgE9Gz4sxJToFJCIQD6ZZukwZePTphuT1m2FjQ66V2L6ZP7X0Rn023J5XY4qrbWCBWz9hoRSOHj2Gd9551yHGFG/nuedf4H+EIKKzv/4GnTMEhxUe0D9EwMEEROOh4nrGxMQIS2KvXbuGkaNGsTgo/YX/jPbcHJV7qFYPWgE+QZWnip2rILBOkHDujv9wY6rV74NQSGPA391/EP1G0Ycj2IpD2svPHV/VPOtMHqo8I8tUwLR4OVQsLIGyS9tM9/kFD7ZzLcGKEC/R/CZ/Qka6QQScTUBU/xt8fX2hZvMkeNLptBCu2bi7vZMhwIDaL9QTjsIViti7eknV99vFTYizJOOXtl+K3pjiYP3ZUF9Vf5r6J6mXjISVJAGPvj2QtG4TWCCpJ+SPTrFi9MEETD2WxIL8ulUowydY0A0ikEZAVAZVmlCO/qvSeKBQaT/h0Hi7d4DIQVX7YmOHmQg0FHU0drvUzyfGdi5lfyPbLsJRJURARgQUBh08G9WFZfOOJ7Ty91QKw4AV2HDg6/+kxq56IhPdIAJuRoB+6suow1cuX4H4hHi2NkqBkkEl0bRZs0xBER9XddaxHxGV/ADjGrz7+CO6JgJEgAjAo193JLz3OQwdWoFFW81EhM9N7xGqQfcQDY+zSYkIuD2BzP9D3B6HtAHMnDEDPCpxAjOqtmzegr7/652tQnwrmTln10HvkXlH92wLiOTBtXgrtt1MDUAoEpFIDCIgXwJeBng2rA3Llp3Z6pjRmLqbZMNXJ2gYMFtY9EDWBCRhUPHtDPhB6ekEXnn1Vbz97rvgMVpiY2MRwSIUZ5XCYq6giMYbg9mQn5RSRJIVRyLoXZBSn5Gs0ibg0b87jKv/TJ2J/hRViuoUKOujQv/tcdh03fSU3PSYCMiLgPiG/B7+3LGyfbjS0siRI+Dr54vRn36adov+ZkPg56U/Qcsm9F+5coUF06uJIkWenHQfb05EJf8y2NdrqSQmomejKt0mAkTAGQS8vKBpUEvwUqk6tMyxxbRhwNaBaqy4nAwb1GwCAiUi4B4EROeh0nqmThLnu86nJb4NCqXcEbCy3ditZqsQs4dHJL569eoTBb879Starx8Es016np6Kfio8X0HzhE50gwgQAccRUPfvwbxUf+TKS8Wl8NYoMKiSNt2Y4p/gPCgvJSIgZwKiM6iKPdxPatnPP6NS+Qr4cfFigf+ZU6fRjQVqq1u7NubPmSfnPimQbi++8jLeGjYEY1nU+bp16+L3desy1RdriseCc+vRPbQ51ErxOSgzCZvFhReLLBioT91mIovHdIsIEAFHEPBmXqq6NWDd+k++ar/FYlb1o2HAfLGjQtIhIDqD6pPRoxEUlBpo02g04uKFiwLNs2fPYhzbJHXatGmYPn0ajh09Kh3KTpSUb9HBh/sOHzqMf/fvR/XqmbePMVvM6Fe6Bd6Q2NwpJyKkpogAEciCgHpgLyStyr2XKmMVJQ1KLGvljcMRJry5O54NBVIiAvIjIDoXRfny5bFn/z5E3b8PblAVDwzExx9+hGYtmgu7i/MuqMA8V3v37Em/ll+35E+juvXqYcqkyUJhn0KF8NbQIWjRqpVwzbf2OX72BG4WisGY+m+zDYZFZ0vnSmkeKX3HbRM+qi6t1Ym5Uo4y2ZVASnwK9s3cBY0hdYi4RI1AVH+utl3bcKvKuJeqdnWYFq+A+pX+yGusBO5d/rS2HtHJ1vShQLfiR8rKnoDoDKo04v6FC6edCn99C/mmX3uwwJwBAQHp13SSSmDOvLlZolizeg3GfP45kuvocK+ZLwKi9WjTMOfJpVlWJJKbticDN4tEMhJDTATUeg2q9KoGv9DUzwoPT9F+3IkJW46yeLw+EJYfVyLh3c9g+HgYEFQ8x/xZPfRjQUHTUqTRigNs1W7HYArWm8aE/kqXwKM3W+Q67P7nH9y6cQN86O9C2AV07Nw53xKbk024fy5COIwPkvJdj1QKTpwwAeeuXsCt+t5QbruJKaMnSkX0J+TUsl+5RfWSeW2fkJ9uOI8Ad8JqffXQFzYIh7vvimAP8gq2NZjH689B//arSJgwA+Zf1+d6onpW7Xuy/898GPDlnXG4FEe/lLJiRPekQ0BS30ytW7VG106d8fIrr6AQG9LKb0qKTsKZdaeEI/pyVH6rkUQ5vkIyJSUFpsQUvLkqDh47biE6Sro6V2Wr/F6u4N7bBUnixSMhZU1AUb4M9HMmAXEJgrfKdv1WvvRNGwb8pKYeS87J/8dtviBRIckQkIQPfPqMrwSgCXFsMiMLauLF4qIUJHkX90GzkdId8sqL7goW16tOnTq4F3EPZ87fxZtFykHVo2teqqC8RIAIEIEnCAjeqtcGQnXhKpImzYK6WSPw8Ap5nVvFK67I9gQcV8/wRBt0gwhIiYCkPFQGNimyoMaUlDrHXrJ+/8NCfD37axRr1wqvV62FT995z15VO72ee2xri6ORZqe3Sw0SASKQNQFF+VDovmHeqiQj81Z9Ctv1m1lnzMPdSw+sqcOA7C8lIiAVApIyqKQCVWxyqtVq9BvQH4uW/IiSg56Hhc97kGi6Hm/BVtrSQqK9R2LLlQD3VqlfHQD9e4OZt2o2TMvXFmhuVdlCSvBhwLFHEoS9AeXKjfSSFwEyqOTVn0/VRtmmGVIOHWdzH+KfmpcyEAEiQATyQkBRJiTVW2VMQeI7zFt1Lf/eKj4MuLSlN9oFpYa9yIsclJcIuIIAGVSuoO7KNtmcKm2frjCtyBxB3ZUi5aXtUG8lOpSiD9i8MKO8RMCZBARv1Sv9oB8+CEmTv4Z52ZoCeaue8X+0M0Jcig18OJASERAjATKoxNgrDpZJ1bY5Ug6wSPPx0vNSBWiVqJ7hA9bBqKh6IkAE8kugdCj0cyfDZjIjcdho2MJv5Lem9HJxbJ9SPgw45XgiEk0Ubz0dDJ2IggAZVKLoBicLwb1UvTrDvEK6c6mcTIyaIwJEID8ElEqoX+4L/YdvImnKHJh/XlUgbxXfx5MPA1Zmw4EjDybmRyIqQwQcRoAMKoehFXfFqvYtkPzvEealShC3oI9JdyragsXnkx+7S5dEgAiImkDpUkLcKpvVxrxVnwBXrxdI3G4hnvj6WQqzUCCIVNjuBMigsjtSiVTIfjlqe3WEeaW05lIlm22ISKI5FBJ5y0hMIvCIAPdWvdgH+o+GInHaXJh/Yt4qq33+L5+KstAw4CPSdOYiAmRQuQi8GJpVdWiF5P1HYEuQToRiNloJD3prxfD6kAxEIH8EQoOh53Gr2P/lxGGjgCvX8ldPhlLV2LxKPgw44O84bLxmyvCETomA8wjQV5PzWIuvJfaLUdejg6S8VLULe2B4dZ34WJJEBSYQGRmJ+XPmFbgeqkACBNhnj8fzvaEbwbxV0+fBtPTXAnur+DDgslbeMNPu6RJ4AeQpolsaVNHhUVg7eKVwXNl5SZ49m0utlJ1aI2XfQdh4bCpKRMBFBH5auhQtm7fAL78ss5sEfL7Og+sPEHUpSjgS7klvVavdYIi0IkUI81bNmQyFSvXQW3W1QJLyvQG5YZWWjBY2Z4tWA6bhoL8OJqBgm+e61drTlLhkbP18EzrPZHtOURII2KIfIPmr+VD4eEMz7FUo9OL1APE4NDHsAzLY4Ja/BWT7xtapURORbNPu4OBg7N631y56bhu9CWazBVqf1C/Y4lVKoFLPanapmyqxPwHzhr9gYj/sdGM/slvlZ9kilpEHEzC4kg6dS6ntVi9VRASyIiCJzZGzEpzu2Y+Awq8QtBNGwrJ9N4tuPBq6N16Esl5N+zVgx5rCHliw/aYJH9cUr9FnR3WpqgIQUBvUqNW3HgLKFS5ALVTUWQRStu9h4RWG2LW5yn4qYRhw9okk/HPbhKkN9HatnyojAhkJkEGVkYabn6taN4W+dvVUb9WOPdC8/ZqovVVu3l2kPhGQDQHbkZNQFvEHgorbXSc+DDi6th58+I8SEXAkARo3cSRdCdad5q3yYB6qhLdHwXqAxaoSUSqqV6JOEXLdi6hLCiTK4kWLMfy995GQmBqkMTo6Wrjm925eL3hk7QIJR4WdRsC45FdoXxng0Pa0Kras8GHiQ4F/0mrANBz0104EyKCyE0i5VaNq1QSGr8Yi5Y9tSP7yG9GEVuBzp1qXJMeqXN63Pbt3Yc3q1TAajYJK8Ww7JH7Nj7PnzslFTdIjBwK2wyegLF4ECCyWQy77PgrxUuJ4pAkv74zDRdob0L5w3bg2MqjcuPOfprrC1wfacR/Do0FtJLzDvFX/sv3/KBEBOxJo36EDevTqCa1WK9RqMBiEa36vZFBJO7ZEVYmVQCLzTnm+3N+p4unVCoyqpReGAjffoJ0XnApfxo3RT30Zd669VFO1bAyvOs8g6asFUP69G5p3BkFhcM2k8EtxVpyOMmVaGm0vPake5xPo268f+FFn5z+Cl8rf3x+zZs92viDUoksIWNmqPg/umSpR1CXtl/dRoXzVR59lfM07Dx5MiQjkhwB5qPJDzR3L+Piw5cwfQ92oXqq3av8hl1CIMVpxLto+21W4RAFqlAgQgXQCfO6U5hXneqfSG8/i5ECEmYYBs+BCt3JHgDxUueNEuR4SULZoBK/a1R56q/bC853XAS/apJReECJABPJGgE8hUAWVgKIYmz8lktSgqAf8tXqMO5yA+ux8WAbvlUhEJDFETIA8VCLuHNGKxgKA8uB76iYNEP/up7Dtc563qirbs2tQ5UeRkEXLiAQjAkQgRwLGn36Dxslzp3IU6OFDPgy4tKU3umaIuJ6bcpSHCJBBRe9AvgkomzWE18xxSN66E8aJswC2QsvRiS999vOk19bRnMVU/5rVa7Dw++/x4MEDMYlFshSAgJVNGfAIDWLeKfEGXeUrAdPStXiry1YDrvrtN4z4yH7R49N0or/2J/DojbF/3aKt0RiXhLBN54SD7/VFqQAEmLdK+8WH0DRrxLxVn8F68FgBKnt6UTObNEoB+p7OSU455s+diwnjxuPevXtyUsutdTH+vAqal/pKhgEPCTr+aAImHU10+t6AIaGhiIiIkAwrdxbULQ0qG4uYm/IgWTgsySZ37n+76a5s2gBes8YjccESZvE4bhny8ftmfH0yNWaR3YSnikRNILBkSZQqVQp67aPVWKIWmITLkkAiC956+NBhJGzfBY+yoUDRzN4pHoNs9+7dzNGd6unm28zu3bMXkRH3s6zPmTe5t2pJC28846/GjJNJmZq2Wq04deJUngPRXr16FYcOPJouEXX/vqDv49vrJsTFoXmLFpnapAtxEnDLSek6Xz2q9a8hzh6RslTeXtB1bgvz6j/g8dz/pKwJye4iAlbbkys4l/y01EXSULP2IrB40SLM/WaOYCx9ExCKi52a463HKtfr9Zg9YxZCS4di+oyv8O2Cb7Fy+XJs3LL5sZyuu+waomZzqx7t1HDixAkMfWsIoqOj4OGhRrWqVfH9oh+g0z3d+OeG00svvoC58+aiGTOYXnvtdVSvUR2NmzTOpGDjpk3RQv2ozUwP6UJUBJSikoaEkTwBVde2MLJ9AJHsGC+VDwvIV5pNGqUkLwIajUZQKC7uyXl4i35YhCmTv0RkZKS8lHYTba5cvozp06bj8LGjKBIdh7/v3sTsn5fi+PHjmQgolUp8/c1s/PXXX1i08AfM++YbzJ0/L1fGSaaKnHTBDaJBE+bhdGATnA+/ieMnT2DLtq0YP25criQoXbo0Ro8ejU8+GSW83ylst4BPP/vsibJqMqaeYCLWG2RQibVnpCoX+8+v7dSaean+dIgG5X1V6FMm9cvXIQ1QpS4hoGcR0nmKZRPP3xn2NvNmfIMrV64I95YvW4b58+YhKipKuM7tPynxKdg7cyc2f7xBOE4sE9e+lLnVQ+r5fl+3HpF37qBD0SB8HlgeP9y5ilu3bmItW2zweAoMCmJGxacYN3Ys3hoyFFWYxyctPT4UlnbfVX/vMJ3iT+6C5coxKL7cBc8SobjPhu22b9uea5EGvvA8KlSogEU//IC53y5ARuNJbPrmWik3zkgGlRt3vqNU9+jWHsbtu5mXKsVRTVC9MiNQveajIfjf16/HtKnTsGXzlgJpqdZrUO+Vhmg+orVwVOr+6Mu5QBVT4VwRsCUmwfLXTnQ+cRkLS1dHcbUGr1w6hnvGRHADukzZMlnWc+L4CRQqVAj79+1Nf85XeH41fXr6tRhOeFR/FQurHnt4C3TrpiPlznVomI4B7H5uE59Xxj14fLhz/9796cX+278ff27YkH5NJ9IgQAaVNPpJWlJyL1WHVjCtsb+X6laiDfvu0kICab0QT5d2wvgJaNCgAZuH8mha55Url59eMIccCvbppvHVQhegFw6NF8UvywGXXR6lGVFJIyYg6YMxsN27j9Dxn2Cq3oI/jLGIMCfD388PIWyRQZ++T67y27ljB/784w/8sWkjwq9dY56bRYJcfEj4mWeesYuM9qrE09MT7Tt1RKmgYJj2roY3246rNFuRN27ihFw38fmnn6FsubL47oeFmMzKhYeHC2WLFCmCIMaIkrQIPPr0kpbcJK3ICXh0b4/4t0ZA/b/O7FvNfkN0txMs2HPbjGeL0SRNkb8CeRLPwBY0rFz1G1JSUnDxwgX4Mg9F8cDAPNVBmV1DgBtR1j3/IWX7Hthi46BpXA+69wZn2p9v06ZNWLFyBbZs2oyWrVphwHMDn5gbFR0djQ8//BCj2Tyi4OBgTJk+Da++9DKaNmuKAGaEbdq4EXwzbTGlL6dMQfv2HbB06RKUKFECbwwejDLlyuVKRK7PVjZfbOuO7ShatCi6sw3B3xk6DGvWr8OFixdx8+Yt1KxZM1d1USZxECCDShz9ID8pmBGlbd8C5nWb4NG3u/z0I40cQoB7IjLOm3FII1RpgQkIRtSuf5Hy995sjaiMjXCD+bXXXxeOjPcznvPwA0uW/ISqz6QOzTZs2BBrf18Pb2/vjNlEc775egqsNgU6tW2DNuzIa6pUubLgiePGFE98Qvq5c+eEHxV5rYvyi4OAJAyq1atWw8/XF63atBYHNZIiVwQ8enQUvFRe7K+9vFRlfJTw8bSfxytXilAmIkAEkFcjKq/IAgICwI+MqWLFisIlj9EkpnQ3yYbvzhnxM4tNld/EV/llTHwIsUaNR3MJMz6jc2kQEL1B9cagwWxy6maMHT9OGkRJykcEmLfBs00zmNdvgUefro/uF+CMbzvjR1NhCkCQihKB3BNwtBGVe0nEk5NHTR/xXzzG1DZAz8K4UCICaQREb1BdDAtLk5X+SpDA3fo1cff19zF08hhUr10L4yZMQPHixSWoCYlMBNyDQCYjisUF0zxb94k5Uc4msZCFFQhi86rEkLgJNbauARn3+rOnXElJSfjt198wcOBAe1ZLdTmBgOgNKicwoCYcSGDwkLcQeOUqQlKMWPbLL0JwxtVr1+a7xRORFuy+Y8LQqtp810EFiQARyExAjEZURgn79e8vTFTPeM+V544yprhOGg8Nvhg7RlT6upK1lNoWrUHFJ+dNZBFnb7PgaTz9tGQptrEVEUOGDUOjZ58tEOPo8CisHbxSqKPmwLoo3aJsgeqTWmE++dNkMoGP2fPEV1bx5eo8UrG9U9Tt26jIJm4OKFsLG2OjWcyV1GCN+W3HZLUhwcSd7pSIABEoCAGxG1EZdQsJCcl46ZJzvim7irmn1ErHDvOp1CoyplzSwwVvVLQGVRzbEHL37j3pGl5ky0j50at37/R7+T3xC/FH55k98ltc8uWMbIuDzh06sv2yZrCAcjoMYxGJ12343b6raeITYF6xDjN92fLnyNPocmwnvOq0Q4UiqSta8guRf5j5aBz7gZZf2aicYwiYLWahYh5EkVLBCDxhRGUR4qBgLci39JjDiegQpEGLQArZIt9eLphmojWo6tWrh2Nsb6RunbvgGgvwNnLUKPQf0B+Gh1tUFExt9y7No/LOmDUTn4wYCSX7kuIbkdptafJDQyr53yPQshhU5X5diGAWtLHW4cNoVKsk27tqlAD/6H0zahXO++tXPUAFflByDwJ8+43IyNQtZ3zZSl9KeSdARlTemT1egodIsDFPOxlTj5Oh64wE8v6NlrG0g8/5B6j6YeRknU4L+kC1H/BatWujKIvGa7VZwc95OnjwIEJDQ8Gj9OY5PWZIGb6bBjaGCF7TN3PnZKrOzEbsVl1Jxo8XktlKGR1btWf/ocZMDdKFJAl8Omo0jh45IuzvV5gtp/d7bEm9JJVyktBZGVHa9wdDUbxgHmIniS+qZthIH1ZdTsHXzxpEJRcJIz4CojaoxIdLPhL9vX07koxJbO6UWohA3LFTJ/B5CpfZvlJ5MqiyMaRyIuXBRm4m1jNgL9tC5jDzVLUpmfu4UtHJVtxnMWD4JsmU5E1gzerV4Hud6bRajP78MyhoyC/HDheMqH/2I2XnPtj46jw2nEdGVI7IcvWQz5ta2NwrV3kpk3sTIIPKDfs/MjISY74YgyU//wS1ygP92J5aderWZXto/ckm/DfKHZF8GFKPV9w4w/YxcSk2mNnwztO8VZdjrdh+04SPa+oerw48+N8UthXE2bNn2a7tGkGX998bDj7Jk5L0CBw8dAj3Iu6hFDP0VSrqw6x60JbAtn3ZRUZUVmzoHhFwNgEyqJxNXATt8dV9q9auSfdE8e0d9NonDZQsRbWDIZVVvTcTrfjkQAIGV9ahY3DeJ33ylYsvPP8C+vTriynTpsFiseCtN97ErNkz8QHbH4yS9Ajw7UpKsyO/ycbGah5cf8A8W6lDyp7eGhiK5r++/Mph93LGZFjYli/kibI72UwVno22QMfc6aHeNCUhExi6yJaAJA0qi8kiKESeh2z7NccHXl5e4EdaShvi0+t0zGOVjTHjIEMqTYZKbAhvWUtvfHUiCQ2KquCfzbyqEgYVmgamlXr0l8//sjGj6uVXXhFuco/GlGlTcedh2I1HOenMXQiYjWaEbT4LrU9qeJDiVUqgUs9qklc/Zd5iKNjqXBrOc1xXJrLQLKMPJWBu4/xvLeM46ahmsRIQv0H1cN6ElcUfSksjR46ATyEffPb552m36K8dCPRjqyifSA42pDK2x7dx+KyOPv0Wny/1+BBgoF6BQP2Tr23Y+fMILVMmvSw/8WM71PODknsSUBvUqNW3HgLKFZYPALbi0XzhCvTzp8hHJxFqMu5oIgZV0qEE+7yhRARyS+DJb6bclnRSPu3D4JMXL1xIb3HYO28LgSjTb9CJ/QnEx8O0fB1S/jsqhD9IW7Vn/4ayr/GXiyngbvexdXUI0Obsdi9cuDAiIyIyVcaH/aKiotKHNjM9pAsiIEECyT/9Bu1z/5Og5NIRmQcPrlzII19TD6SjJUnqCAI5f0s5osU81lmsROq+b8t+/hmVylfAj4sXY+XyFVhfgO1L8iiCrLMfPnQYH7E5Rn9u2ABhKJUbUt//jIT3voAyKBDckFJ1bCWEQHA2CL69zAsVPDHntFFoOjY2FpMXrsRLM1YIsckyysOj5/PAr3fv3k2/zd+Td4YOS7+mEyIgaQJRMbCcvwRlk/qSVkPswvPgwS9VpB3Yxd5PYpRP9B6qT0aPRtj5MNy4cQM8wvfFCxeFv8kphcTIU1IyLV2yBOPGjGUrqSKwfsWveFC7EboXDUr1SL3GNuZ0wFY0eQXUoKgHm1PlIWyV07F9B1xO0cJatj4OfjsGG9iqxLLlywpV8hhlI1nQ0N49e6Fdh/a4d/ce9u/fh2XLl+e1ScpPBERJwPTTKmgHkndKlJ1DQhEBRkD0Hqry5ctjD/tiPHL0CPaxv+MmjC9wx8Xfj8eBefuE497p1L0CC1ypBCv4dsG3uHL1Kvr4FsN03yDsvX4V6rmTXOaRygnh/v37cTvOiDsf/AajXwlcvX4Ns7+elalIHxb+ga9erF2rDrr16I4dO3eiYsWKmfLQBRGQJAHmnUo5G0beKQd23qJzRpyKSl3w5MBmqGoZExC9hyqNvT+bI2OvpNGrEVgnWKhOFsuo8wlGx1b1jShdBclWC3qd+RdVKleB6mFk+nxWabdifJXNsSgzTkRa8EYVLQoVKgSNzgDvRe/BHHYAWhbskc+bejwVK1YMnbt2fvw2XRMBSRNIWbYauoFs/1EKbuqQfjzJDKmdt014paLWIfVTpe5BQDIGlT27Q6P3RFCDVIPKnvVKra7v6zfDqgfbMC/yOipXqoznnn/eJQEU+frNq3FWsKgHKFso1Wk696xRcJ/WDGBhHFiG6tWro2pQEXicP4JkjQV+Zcpi2NChUkNO8hKBPBOwRT+A6dQ5aIa9mueyVODpBJJYvLIvDqeGSCB79em8KEf2BCRrUNE2FNl3am6eJM/6FkGlQjBo+wbUOngIFStVRHCwc4xM7n3SsRAJfEHyFWZIffRvPEJY8LxepT3TDaqPqj8eaFSBVWtW4/TJ04h5EIN69euBByilRATkTsDEvFPa/uSdclQ/a9neMjMaelGIBEcBdqN6JWlQTZ/xlRt1kf1V5caUUuMJ9ZCX4cuqb9O2jf0byaLGH8OS8ee1ZOjYB9icxl7w0ShYJGwlVrX1ySL3k7e4EV2tuvQDMz6pGd0hAlkTsMXEwnTiDDRDUwPWZp2L7haEAP9hR9HQC0KQyqYRkKRBlSY8/c07AcGYYvOP1G++lPfCuSwRb7ZhMZvgeYzNf6oZoMLb1VK9Td1D1HipvCdNA8klR8pGBMg75bh3gH9OGdjWMhS603GM3a1m0a/yc7cOcaS+jjCmbiZYsepyCkaxffius3Oe+K4x1fw9ML2hPt2Y4vd51HOao8BJUCICuSDA4q6Zjp+GqmXjXGSmLHkhwPZhx5Dd8bj8IPUzKy9lKS8RyI4AGVTZkZHZfXsYU/wX3R62Embz9ZR0OmeizTCxT6cXK2gRpE99nXhgvJaB6ie2jUkv5ICTT0aMdECtVCURcB0B089roOvbjVb2OaALvmWLXp4tpk6fs+mAJqhKNyRAQ35u0On5Mab4yjv+662ITiHMdWK2FN7Zk8A8Tyo0Lv5oA+W2QRpRELx7133jiYmiA0gI+xKIjUPKsVMwvOW4oXn7Ciyd2mJSbDh634wFTR9tEC8d6UlSMRMgD5WYe8cOsuXWmOL7V6Wl9VdT0GdrLL5l86CiklPvs6kGWNTCC8PZ6jseudyZie/JN2H8+PRtZWbPmImrLCBpWrIxD1mPHmwVFCUiIBMCpl/WQtenK3mnHNCfvmwxzIJmXjT9wAFs3b1K534zujttJ+jPt+fhe94VLVoUuTGmxhxKwukYE8r6qPBlfYMgYbdQDbqzQyxJpVKhTGhpjPz4Y7Rp0wanz5zBu8PfTxePr/7rRgZVOg86SSWQEp+CvTN3QmNIDa8RWKMkqj9XW/R4bIlJiP95LXwXTRe9rFIVkCaiS7XnxC03GVTi7p88STd/7jwsmD+fxcG0YWRgOXTv0gWGh6v5wuOtWHc1WXB1v8zmO7Vgc5x4erWSJ4INuky/1sT4YTPwheexddtW8O1y/ti0MU9cKLN7ElDrNajWqzp8S/sLAFSeKkmAUOh18F3wJZKX/gZrQhI8e3eG6tl65K0qYO/9d8+MCixwMF8cQ4kIOIKAW75ZVjaElBSVJByWZHns3XT37l3MnTsHJ8+cRk/44LJZgxc1pdPfGb6qpQZbeff1s17pxhR/WMpLGivvLCYLIiMjmeGnSB/6+4/t7xceHp6uI50QgYwEFOzTTeOrhS5ALxwaL+kEglVUqwjthJHQjRgK66ETSBj8ISxr2Q+J5EcLQjLqSuc5E7idaMOU44lsBbIYfy7mLDs9lQ4BtzSoYm/HYeekbcJx+/B16fRWDpKeP38e8QkJKM72u2vvXxwzLApYD6xPL8ED13GvFA+mKcU0bdpU1K1bD5O+/BLvvf0OuIGVmJiIlBT6gpFif5LMuSOgKF4U6ndfh372BFiNyUgYOhKmb5cCbLNkSrkjwH9MjvgvHmPrGKBnOzRQIgKOIuCWQ36+Qb7oOJ1N+JRRatCgAYoULoKz9yPx752raAAj3vzkFVlomBAXj4TEBHz62WfCdjM9evXEkWNHcO9eBPz8U4dzZKEoKUEEsiHAhwHVA3pC3a87LDv3IfGzqVCWKALPfj2hKB+aTSm6zQncT7ayOaGeeIatUKZEBBxJwC0NKkcCdVXdfF+7Pzb+ialTpuL8hcv4tnAwSnXo4Cpx7NquwdsL4ydMSK9z0ODBwvnlfKGYwQAAIbdJREFUi5fS79EJEXALAkolVK2aQM8O26nzSF6yQphnpevTBYpGdWmeVRYvQRGtEv8rLZ5FNlmISLdkQoAMKpl0JFejWLFi+OrhPofGsTNgY3uAKapXkZGGmVV5pvozCChcOPNNuiICbkIgbZ6V7c49mFf+jpTFK6Dt1AaqTq3ZdgVkQLjJa0BqioiAW86hEhF/h4ni+Wo/JP240mH1i6HiKlWrCkakGGQhGYiAqwhknmdlpHlWDzviy6OJiDTS1jKuei/dsV0yqGTa64rgklD6+8J6/JRMNSS1iAARyEggbZ6V4bvpUJYrLcyzMk6YCdvFqxmzucX5pusmxJhsCGDDfZSIgLMI0JCfs0i7oB3tywOQOH0e9LOquaB1apIIEAGXEODzrFo3hZ4dwjyrH91rnhUPkfD9uSQsa+XtEvzUqPsSIPNdzn0fVBzKIv6wHTkpZy1JNyJABLIhkDbPSv/xUJgPHn0Yz2qTrONZ+WuB2Szenk5FIRKyeS3otoMIkEHlILBiqVb7Sj8Yl/wqFnFIDiJABFxBoASPZzWYeat5PCsj4oeMgOm7n2QZz4oH7ww20FebK14zd2+T3jq5vwGBJaAsXgS2wyfkrinpRwSIwFMIKNg2Uzyeldf3X0FZNjR1ntXEWbKYZxVvfrTB+1Mw0GMi4BACZFA5BKu4KvV8uT8SyUslrk4haYiAKwmkzbOaOwme3dsjmc2zShz+BWz7DgE8tLjEUiKbgP7i33GIS5Ge7BJDTeLmQIAMqhzgyOYRc/d7BBZje4Idl41KpAgRIAL2IaCoVlnYN1D/EZ9ndSR1ntW6zZKaZzX2SBIGV9LCW6Jba9mnJ6kWVxNwS4Pqwc0YbPn4D+G4vv+aq/vAKe1rXmZxqRYsZfswRDqlPWqECLiagM1iQ+y1GERdvC8c8XfiXS2SuNvPMM/KxvbJTJtnZYuJFbXc52Is8FDa0CGYgpmKuqPcQDiFjSU30DNdxZS4ZGwZ9Qdaf5G6LQvfgd5D6x7RI2wnzyBx9g/Q9uwEVWcWTZkSEZAxgW2jN8FstkDr4yloWbxKCVTqSSFEct3lViusO/Yg6dffYfhmIou+nsox1+WdmJF/idGaPicCp6ayJOAelsRjqis9VNAXNjx2V/6XimeqwDB3MkzMU5Xy0TjoRgwFCgfIX3HS0C0JqA1q1OpbDwHlaHuifL0AbJ6Vsk0z6Fhh4/T50I5+L1/VOKMQGVPOoExtPI2AWw75PQ2KrJ+zPb7U774O7Qu9kTByEiwbt8laXVKOCBCBghHgRpVCrYblz+0Fq8jOpXfeMiHZ6lYDLHYmSNXZmwAZVPYmKpH6+KbJBrbCx3rhMpI+Hk9zqyTSbyQmEXAFAc/3BsG4YQtw5aormn+izRORFvwYZoRGQb6pJ+DQDZcRIIPKZehF0DCbE8GD/Xk+1yvVW7VphwiEIhGIABEQHQGNBno25JcweQ4b/0t2qXg8RMKYIwmY0sALZE+5tCuo8ccIkEH1GBB3vFTWqCp4q8znLyBpxAQgMsodMZDORIAI5EBAERwIbb/uME6dl0Muxz8Kj7fi7ao6FNORd8rxtKmFvBAggyovtOScl3mrPN97A54DeyBhxERYyFsl594m3YhAvgjwTZeVOk+YN/yVr/L2KFTZT4WWgWp7VEV1EAG7EiCDyq44pV+Zska1VG/VOfJWSb83SQMiYH8CGjafKoUtZrFdDrd/5VQjEZAwATKoJNx5DhOde6veZ96q/txbNQGWzX87rCmqmAgQAYkRYCv+9KOHI3Hy10Ci0SnC82iJnx9MAG3X5xTc1Eg+CZBBlU9w7lBMWYt5q+ZMhuXMeSR9wgL7RcW4g9qkIxEgAk8jEFQcWrbJsnEam6TuhDTvjBFBXip40LQpJ9CmJvJLgAyq/JJzl3JaT2iGvwltn25I+HgcrH/tdBfNSU8iQARyIKBq1QRKby9Yfmf7/jkw8RAJByNMGMT26qNEBMRMwC0NqpTEFNw6fEM4EiMSxNw/opFNUfsZwVtlPnE2dSUgeatE0zckCBFwFQHN268hedPfsF246jARgr0UmNbQQCESHEaYKrYXATc1qJJxfX+4cCTcjbMXS/nXw71Vg5+H6cwF2K5el7++pCERIAI5E+DzqT5j86mmfgNbYlLOefP51M9TiSJat/yqyicxKuYqAm65l59XYW80GNbYVcyl225sHBJHjIf3mA/APVaUiAARIAIILAbdc71hnDIHurEf2Q0ID+CpV9OkKbsBpYocToDMfocjlkkDD40p3evPQ1GnukyUIjWIABGwBwFli0ZQ+vnAsnaTParDrUQLXtkVD9qpzy44qRInEXBLD5WT2MqnGWZMJbD9/vSDyJiST6fKX5OU+BTsnbkTGoOnoGxgjZKo/lxt+SvuIg01w15DXK/X4dOAMWZeq/wmHiJh5H+J+Ky2HuSfyi9FKucKAmRQuYK6lNokY0pKvUWyZiCg1mtQrVd1+Jb2F+6qPFUZntKpvQmYF6+Atml9oETRAlW96boJTYqrUY1FRKdEBKREgAwqKfWWs2UlY8rZxKk9OxJQsAkNGl8tdAF6O9ZKVWVFwPT9z7DFxELz4Vso6HK8TqXUbKiPtpbJijPdEzcBmkMl7v5xnXRkTLmOPbVMBCREwJ7GVJraNNSXRoL+SokAGVRS6i1nyUrGlLNIUztEQNIE7GlMbb9pljQLEp4IkEFF70BmAmRMZeZBV0SACGRJwJ7G1J/XTNhxKznLdugmEZAKAZpDJZWecoacZEw5gzK1QQQkT8CexhQPkfDD+SQsa+UteS6kgHsTIA+Ve/f/I+3JmHrEgs6IABHIloA9jSneyNloG8bVNUCnoplT2UKnB5IgQB4qSXSTg4UkY8rBgKl6IiAPAoIx9SDOLqv50oi0LklfQ2ks6K+0CbjlmxwdHoW1g1cKPVdzYF2UblFW2r1YQOl50E5N5QoUAb2AHKk4EZAzgXRj6oM3CxwaQc6cSDf3JeCWQ35+If7o+V0/4XB3Y4q/+obpX8CakICkL6YBsbHu+7+BNCcCRCBLAvY2pvg+fROOJGbZFt0kAlIl4JYGlVQ7y2FyexngOepdeHZshfj3mXG1c7/DmqKKiQARkBYBextTXPsxR5LQoKhbDpBIq/NJ2jwRIIMqT7jknVnZsA4MX0+Caf9BGD+fSt4qeXc3aUcEnkrAEcYUD5GgVtrQNkjz1PYpAxGQEgEyqKTUW06QVWHQwfOTd6Dp1FbwVll27HFCq9QEESACYiPgCGOK61jNX4VP2cbHlIiA3AiQQSW3HrWTPsqGtQRvlfnAURg/myLs02WnqqkaIkAERE7AUcYUVzvES0khEkTe/yRe/giQQZU/bm5RSvBWjXwbmq5tkfDBFyBvlVt0u2yUtFlsiL0Wg6iL94Uj/k68bHRzpCKOMqaSWH9QIgJyJkAGlZx71066KevXhuGbSTAfPAbjp1/CFv3ATjVTNUTAcQTMRjPCtpzDyZXHhOPGf+GOa0wuNaekIHHjDqjfG2TX0AhHI834+D9a1SeX14T0yJoALbPImgvdfYyAQs/mVo0YBuuh40j8aCx0/XtA2abZY7nokgiIh4DaoEatvvUQUK6weIQSuyQaDfQ9O8L6xzaoenSwi7Q8RMK4w4n4tiltLWMXoFSJaAmQh0q0XSNOwZR1a6SuBDxyEsbRk4GoGHEKSlIRASKQLwLq3l1h/OMv2MzmfJV/vNAP540YUkWLojraWuZxNnQtLwJkUMmrP52jjV4Lz4+HwrNnJyR8PA7Wbbuc0y61QgSIgOMJsP/fmiYNYd243S5tDauqoxAJdiFJlYidABlUYu8hEcuneOitMh89BeOoSeStEnFfkWhEIC8EPPoyL9XvW+zipVKQYyov6CmvhAm4pUGVFJOI078dF47oy5ES7j4RiM5/zX40BJ7/64L4ISPZ3IutIhCKRCACRKAgBPicSU3jerBs3pGvaviCvl13TPkqS4WIgFQJuKVBpfRQQuuvFw6VJ83Lt8fLa30QC5taDYW/rz2qozqIABFwMQF1v54wrt2ULy/V7JNJuBprdbEG1DwRcC4Bt7QmPL20KNu6vHNJy7Q1HkIh+av5UHh7wevbqeC/bCkRASIgAwJ8rmTzRmwuFVvx1y33K/4ORphxNsaC75rRZ4EM3gJSIQ8E3NJDlQc+lDUHApbtu4UQCpruHYSQCmRM5QCLHhEBCRJQ9+7G5lLlbcVfeLwVX9bXg6ZOSbDDSeQCEXBLD1WBiFFhIbCn4JXy8Yb+64nklaJ3ggjIlQCfI9m4ASybdsCja7tcadm7NG16nCtQlEl2BMhDJbsudaxCPEQCD+zp2aOjEDqBvFKO5U21EwFXE/Do1w3J6zcDVpoT5eq+oPbFTYAMKnH3j3ikYwE8eSBPHiLB8PUk8JAJlIgAEZA/AWHF37P1coxLxYf5Zp9Kkj8M0pAI5ECADKoc4NCjVALWrf8IATw9e3UWQiSADQNQIgJEwH0IePTrgaR1WXupzCxEwsj/EtApiIb63OeNIE2zIkBzqLKiQvdSCTCvVNL0eVD5+8Ewh20zo/UkMkSACLghAYWB7eXZqLYQl0rVqU0mAt8wz1THYA3K+6oy3acLIuBuBMhD5W49nkt9LX/tFLxSWravl+bDt8iYyiU3ykYExEzAYrHgx8WLce7cuTyLyb1UxjWbnphL1S3UEy9UoB9beQZKBWRHQJQeqlEjP0HRIkXw3gfDZQdc9ApFRjGv1HyoigSQV0r0nUUC5kQgOT4F/0zdDo0udSgqqHYwar5UN6cisn+2betWTJ86DbVq18ZPy37Om75eBqirlId170EomzZIL1vWm36Xp8OgE7cmIDqDqmG9+rhz5w66dO3i1h3jCuUtm/+GcdUG6Ia+CmWtaq4QgdokAnYj4GnQoEb/5vAvGyDUyXdIcPe0YvlyvPHmG5g7dx5uXr+BksFBuUdis8F0/hLUrw4UysSzyVNeHhRtKvcAKafcCYjuEyYqKsopzK1mK/gBNqHS7dP9SCSNmADz2TAY5k4mY8rtXwjHA9i8aRMOHTjk2IbYd72H3gNqg1o4VJ7uPccnMuI+9u/bjz79+uHZZxvhxyU/5om/Zec+qKtVhsLXB6uvpOC7M8Y8lafMREDuBERnUDkDePS1KPw+bJVwhP9zyRlNirYNm9mMeGZMeQ7sAc/33wA8aS6EaDtLBoLdunEDLw58Dm8OfgNXrl6WgUbSUWHFiuWoVasWihcvjn79+2PtqtWwmCy5U4B5p4wr1kH9/P/AQySsuGTE0Gq02jd38CiXuxAQjUG1edNm1K5REyZT6g7lW//aKlz/sHCh3fvCr5Q/eizoKxwhLcravX4pVWjduB2eTRtBWYOG+KTUb1KVdfPmLdi1e7dUxZe03KtWrUL9+vVx9uxZBJUMhjE5Gb9vWJ8rnay790NdtSIUfoUw/0wSJtU1wFNJw325gkeZ3IaAaOZQ3Yu4h4zDfcnsPzs/rly54jad4WxFuXfK+PsWYfuYrNreu2cvLFYLmjVrltVjukcEiIBECPDh1Vs3b2Hzli3CwcX28vLCsp+WoWevXjlrwbxTSb+sg2HSKCHf5PoG2qcvZ2L01E0JiMagevHFF9G8eXO0adlK8FK1bNUKY8aNRdGiRd20axyvtmXzDmGfruy2j5k8cSKMRiO2bNsKlcq95584vjeoBSLgOAK//LIM7Tt0wNdzvklvJCwsDB3atgP/W6FChfT7j59Y9xyApnI5wN9XeER+qccJ0TURSCUgmiE/Lk5ISAgUitT/rgaDXrjW6XTUV44gwPblSmaRj/k+XVml/fv2oRiba1G1WjVs3sgiJFMiAgUgwFfuDn/vfaxfty69lpXLVwr3Fn7/ffo9OrE/gaSkJGzZvBm9e/8vU+XciKparSqWLP4x0/3HL4y/rEbigD7Yf8/8+CO6JgJEIAMBURlUGeSiUwcTsGz6Gxq2P1d23qnlvyxHj1490btvH/y0dImDpaHq5U4gOjoaa1avxvHjx9NVPXTooHBv3Zq16ffoxP4E7t69i1dffw1Nshi6/2TUKISEhmTbKPdOqSqWxWcXNUjhe8xQIgJEIFsCZFBli0bGD5h3yrh2I/NO9chSyYS4eOzetQt+vn7QeHgIk1hpLluWqOhmLgn4+voKBnr1GtXTS9SuU0e416ZNm/R7dGJ/AqGhofjgww+hVD75cf9s48YY/AZb3ZtNMi5bjT9b9UFRrQLNA9XZ5KLbRIAIcAKimUNF3eE8ApYtO4V9ufj+XFml1WtWI7hUMBsmYNtMsFSpYkVhu4qx48ZllZ3uEYGnEihRogRmzZ6NRQt/wInjJ4T8AwYOQJ++fZ9aljK4hoBt30F4lC+NGI0eH1WjcCqu6QVqVUoEyKCSUm/ZQ1YeT2b1nzDMyt44+u3XX/HFF2NRt37qNh3Xr19H9y5dMWr0aBamij5Y7dENVAcREDuBpJ9XQz/uI7xamOJNib2vSD5xEHjSB+xiudImpVusNF7viK7g3ilNg1pszbQhy+r5XJfg4FLpxhTPFBwcjM5dOuPUyVNZlqGbRIAIyIuAdf8heJQJAQqnbtsjL+1IGyLgGAKi81CljfNfunjRMRq7Ya08WOpvK1ey1XobMdGsR9APM7Ol4Ofnh3kL5j/xfDwLoUCJCBAB+RLg8yRnzZiJ06dPY1j55kgaNBgD5KsuaUYE7E5AdB4qHmyOp7Dz5xHKPCVD33oL+9gS/hrVnkHpUiEoExKKtq1a2x2EnCscPGgQ3h8+HNE79+L7U8cwic1loSROAjHhMVjSbSH2zfwHtw5ch9lIS9XF2VPykurq1avo2rkzflj0AxLDrmBmmeb4edK78lKStMkzAVOCCatfW4FjSw4h6nwEbGzKCKXsCYjOQzVl6hQMZr+MzCyKN08XL17C6BEj4anV4r8d29n+U2swefIk7GbbVzRt2jR7zdiT3VN3wJKSea8qq8mK+5fuY+eErTmWldNDzxMeaOL1LBp5F8GRhEj4b7iJnXAf/aXUl8mxyUiMjsfhFYdwbPVR2NjQt9ViReX2VdBmXEcpqeJ4Wdlne8TpO9m2kxSdhOgLkbCSUZoto7QH82fORfK9ZASoA1D2pWk4vWY2LHG38e/6fShbtkxaNvrrZgSSmUF1/1IEbp++Bc1PnlCqlNAX0qF41RIIqlMK5dpXhFItOr+My3pJdAZVK7aE+uKVy7h04QIexMbimerVhSjdX8+chXeHDsPlh1vRREREPBVa9T61YLVZM+UzJ5oQe/cBagysnem+nC/Grh6P85EXse++VVg6XadkbbfSX0p9G3stBhf3XoDOW8c+qFTwD/ETPrgqdKosJTWcIiv/tRy2+Vy2bcXciEH4/ivQnqZJ1dlCevhAEW5DFZ/KsDID3rb9T5S8mwwfrxBc/ecyLBdSnlacnsuUAN88mzul+P+15DijYFCZEpMRGR6JW8dvQRdgQHCjUjLVPu9qic6gSlOhbPnyaado1aw5rrGVZnz5foeOHbHkxx/ZL/fMhlJ65gwnhUr7ZbhKPU2JS4Zap4FfGfeZbDl01NsYN3YsUlJM0Om0+GzqZ26l/xMvgYhvaNmvv5bvt0JQo1D4lCwkYkkLLhr3vBUkKdjmvI0/bJFtFf9M2oZqfWsioFzhbPPQg1QCEWVj8NvwNbgcfgUIh/DDq8Yz1bFw4mL2mZF1eBViJ38CybFGHF93DN5FfFCkfFGUb12eGVChzJDSy1/5fGgoWoMqoy7cmOrExvf5PlSTJ00SHiUbkzNmofMcCPTu00cwRC9duoRKlSpR6IMcWLn6Ef+gqtK7hqvFcFj72gxhN2IexDisHao4bwR69OzB9vQ7jzWr2DYzSYkILBGIiZMnkzGVN4yyy63Wa/DSmtdl/+POXh0nCYOqTbt2+GPDBmz880/BGPBg0bsPHzmMgc8/Zy8Osq+HT/avUUO+X9Sy70CZKOjt452uyZJFi2FMMrIN0IthwHOp68kusKH+wwcPIib6AUJKh6Bd2/ZQsaFPSo4n8PGIEfhg+Ie4H3UfxYoVc3yD1ILoCSg9lGRM5aGXJGFQLfh2AfgGn7dv3kSZcuXyoB5lJQJEQEwEatSqBf6DiC86uXX7NmbOmAEfHx/BoPp2wQJM/XIKeFR1Hj7l2rVraFC/PlauXiUmFWQtCzdeyZiSdReTcg4kIJnp+Xwcn4wpB74JVDURcAKBUqVK4Uu2krdwwKM5jLFs8cn9+/ex+IdF6NipE/bs34dde/dg8Jtv4L8DBxAZGekEyagJIkAEiEDBCCjY7H33CizBtOUrF1QaGkYo2KtDpYlAwQjcZh6qmJgYIRI/H5LmH0W3bt3C2TNncfnyJezYth3//vsv9rI4dCWDg/LcGE1KzzMyKkAEiEABCEhiyK8A+j1ZVAEypp6kQneIgNMJ8KE9fqSlCePGCyt4PdRqlGND+76FfNIe5esvXwXI/rtTIgJEgAg4hYBkhvycQoMaIQJEwCUE/t/evQVVdd1xHP+hiGkbNUgYH0zLgyCgxomPZsZOrUUNSKJ9KIZoWhNHEwS1Jq0gBrEmTVtDRJsoJmJjk+hLqgioaARqBc10rKIjeMmDOoPpOKmDTtTIRexe23o8hEtppHs8a3/3DLjOvpx1/p/lMP+z97qYZU+KN23Si3Pnqv5Ug8p379LESXdWRGhr7zg5b28/4A+zJ2koUyb0lovzEEDgPgX8d4fqPsG4HAEE+l7gIWclBNNZvbm52f05ceKE1r+73q3IDEhhQwABBB50Ae5QPegtxOdDwAcC5tGfuTu1/ZNPNCo+QXPnvKC0mWnuaL+jR/7hAwFCRACBUBfwX6f0UG8xPj8CFguYASONXzTKjAYMC6MH1P+7qY13dvZS3bp157HqUynJSkpK6lTtX5wJP2tratz9sbEjlJGZ2ekcdiDgdwEe+fn9fwDxI/AACZh5kGJiYh6gT2T3RzHej0Q+ovc3vucGWrl/v6qqqxUVfW+5nlMNDVqWne0+ijUnrd9QZDcK0SHwLQV45Pct4bgMAQQQsEEge2mORiXeWXz76tWrWurMmH53M33aMjMWBJKpmenPKnla8t3D/IsAAkECJFRBGBQRQAABvwmYu1R/dAYAfMcZGGC2/Z9+qrKdO93yyhX5MmuAmi1u5Ej9ZtUqt8wvBBDoLEBC1dmEPQgggICvBEbEjdDSZTmBmPPzV6pkxw5t27rV3feQs1LF+qINioiICJxDAQEEOgrQKb2jR8i/amlp0Uvz5qnV+TduZLzy8ld0iKno3Q2qrT0ohfXT24VrFB0d3eE4LxBAwL8Cz8+arb8dONAJ4E1njcW7C1h3OsgOBBBwBbhDZdl/BPMNMnHUKB08WKPNxcUqLy0NRHiotlarV//BPRYXF0cyFZChgAACRmDNmjWKGjq0A0ZySgrJVAcRXiDQtQAJVdcuIb13ySuv6PGxj7sx5L2Wp6amJpkFaJcs/qU7PHrMmDHKzV0e0jHy4RFAoO8FzOi+5GkpHd44LS2tw2teIIBA1wI88uvaJeT3nj9/XslTpurGjRt6+pln1NLSrIo9FRo0aJB2Vexx5/kJ+SAJAAEE+lTg9OnTmp76tG7evBl43+GPDddep6O6WcCaDQEEuhcgoereJuSPbPt4m3KcSfuCt8J1azV9xozgXZQRQAABtba2KuWpZJ09c8bVMAnUtWvX3PL0n85Q4dq1KCGAQA8CPPLrASfUD5lOpD+aODEQxrTUVJKpgAYFBBAIFvjt628EkqnRTj/Mvfv2Be5KlWzfobKysuDTKSOAwDcESKi+AWLTS7OcxL++/DIQ0hnnm6f5FsqGAAIIBAvUOINYtnzwgbvLDGwpWFuo4d9/TDlBUyksz1mmS5cuBV9GGQEEggRIqIIwbCsWrH5LJ0+eVHh4uPvz+dmzemPV67aFSTwIIHAfAmbAyqtLlqi9vd19l4yMDCUkJLjl52bP1vjx492ymUV9cdbC+6iJSxGwW4A+VJa275G/H9HMtJ+pra1NWYsWKmLAABW8VaB+/fppy0cfasKECZZGTlgIIPC/CLw8f7727N7jXpKYkKjyit3q379/4C2+aGxU0k+SdP36dXffsuW5mudcc/v2bdU4CyafcTqy9+8frnHjntAT48YFrqOAgN8ESKgsbHHTkXRq0mQ1On8IzTdNM6ovLCxMqckpqq+v17Bhw7Svcr+GDBliYfSEhAACvRXY5/STmvfiXPf0Ac6XrtJd5Ur8z7p+we/x5y1blLf8NXfXwIEDVXXgr1qYmaXjdXUaERurf168qK+cvzt5+fma88Kc4EspI+AbARIqC5t68aJFMp1IzaO+0vIyjRo92o3S9KEySZWZTX3K1Kna+P57unDhgpa++is1nDqlcOdbqdn/5u9/Z6EKISGAQF8JHD50SLPSn9OGjUWaPGWKO7/d5B9P0pDISG0v2dFX1fA+CISUAH2oQqq5/vuHra6sdJMpc+b8l18KJFPmdXx8vBZkZpqi9lZUuKN21hWu1cPO3FSHPjusde+8467dde7cOfccfiGAAAJdCYx/8kmd/vysfhATo9KSEr1dUKAbztxVzV9/3dXp7EPAFwLcofJFM3cfpOkHUV1ZpRPHj6u+oUHVVVXavrNEY8eO7f4ijiCAgK8Frly5olnPpruDXh6NinKWu0p0uhhc1HedRZR37a3wtQ3B+1eAO1T+bXs38l87j/tyc3N11Rnpk5o6zd1nkiw2BBBAoDuBP20qlplVvaR0p47UHdOHW7cq0lkDsL2dvx3dmbHffgESKvvbuMcIPzt8WBkLMrRiZb4inM6mZlRgW2tbj9dwEAEE/C3wvUEPu9MsmKWtrn91TRuLinTs6FHdbLm3ZI2/hYjejwI88vNjqwfFvLl4s9YVFrpr/EUOjdTly5eVlZWlmenpQWdRRAABBO4JmCTqFz9/XseO1blfwmKdkX5jnAXZy0vLdPR4nQYPHnzvZEoI+ESAhMonDd1TmGb2dDO5X5TTF4INAQQQ6K1AU1OT2ttuKSr60d5ewnkIWCtAQmVt0xIYAggggAACCHglQB8qr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFoBEiprm5bAEEAAAQQQQMArARIqr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFoBEiprm5bAEEAAAQQQQMArARIqr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFoBEiprm5bAEEAAAQQQQMArARIqr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFoBEiprm5bAEEAAAQQQQMArARIqr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFoBEiprm5bAEEAAAQQQQMArARIqr6SpBwEEEEAAAQSsFSChsrZpCQwBBBBAAAEEvBIgofJKmnoQQAABBBBAwFqBfwNn1B88CdqdbwAAAABJRU5ErkJggg=="
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "费曼路径积分\n",
    "\n",
    "\n",
    "问题：\n",
    "\n",
    "众所周知，一个附着在线性弹簧上的经典粒子会经历简谐运动，其空间位置是时间的函数，由$x(t)=A \\sin \\left(\\omega_0 t+\\phi\\right)$给出。问题是，如何采用这个经典时空轨迹$x(t)$，并使用它来为束缚在谐振子势中的粒子生成量子波函数$\\psi(x, t)$。\n",
    "\n",
    "理论(费曼时空传播子)\n",
    "\n",
    "费曼寻找一种量子力学的公式，它比薛定谔理论更直接地与经典力学联系起来，并且从一开始就使量子力学的统计性质显而易见。他遵循狄拉克的建议，即哈密顿最小作用原理可以用于推导经典力学，这可能是量子最小作用原理的$\\hbar \\rightarrow 0$极限。看到哈密顿原理处理的是粒子在时空中的路径，费曼假设描述自由粒子从时空点$a=\\left(x_a, t_a\\right)$传播到点$b=\\left(x_b, t_b\\right)$的量子波函数可以表示为\n",
    "$$\n",
    "\\psi\\left(x_b, t_b\\right)=\\int d x_a G\\left(x_b, t_b ; x_a, t_a\\right) \\psi\\left(x_a, t_a\\right), \\tag{39}\n",
    "$$\n",
    "其中 $G$ 是格林函数或传播子\n",
    "$$\n",
    "G\\left(x_b, t_b ; x_a, t_a\\right) \\equiv G(b, a)=\\sqrt{\\frac{m}{2 \\pi i\\left(t_b-t_a\\right)}} \\exp \\left[i \\frac{m\\left(x_b-x_a\\right)^2}{2\\left(t_b-t_a\\right)}\\right] .\n",
    "$$\n",
    "方程是惠更斯小波原理的一种形式，其中波阵面$\\psi\\left(x_a, t_a\\right)$上的每个点发射一个球面小波$G(b ; a)$，它在空间和时间上向前传播。它指出，通过所有其他小波的叠加和干涉，产生一个新的波阵面$\\psi\\left(x_b, t_b\\right)$。\n",
    "\n",
    "费曼认为，另一种解释（39）是哈密顿原理的一种形式，其中粒子在B处的概率振幅（波函数$\\psi$）等于从时间A开始到时间B结束的通过时空的所有路径的总和。\n",
    "![sec3-15.7.png](attachment:sec3-15.7.png)\n",
    "这种观点通过沿不同路径具有不同概率来体现量子力学的统计性质。所有路径都是可能的，但有些路径比其他路径更有可能。而路径，在薛定谔的理论中是一个完全经典的概念。路径概率的值来自哈密顿的经典最小作用原理：物理粒子从时间 $t_a$ 到 $t_b$ 沿着经典轨迹 $\\bar{x}(t)$ 移动的最一般运动是沿着一条路径，使得作用量 $S[\\bar{x}(t)]$ 为极值：\n",
    "$$\n",
    "\\delta S[\\bar{x}(t)]=S[\\bar{x}(t)+\\delta x(t)]-S[\\bar{x}(t)]=0\n",
    "$$\n",
    "路径被约束为通过端点：\n",
    "$$\n",
    "\\delta \\left(x_a\\right)=\\delta\\left(x_b\\right)=0\n",
    "$$\n",
    "\n",
    "这种基于变分法的经典力学的公式，如果将作用量S视为拉格朗日沿路径的线积分，则等价于牛顿的微分方程：\n",
    "$$\n",
    "S[\\bar{x}(t)]=\\int_{t_a}^{t_b} dt L[x(t), \\dot{x}(t)], L=T[x, \\dot{x}]-V[x] .\n",
    "$$\n",
    "这里的$T$是动能，$V$是势能，$ \\dot{x} = d x / d t $，方括号表示函数$ x ( t ) $和$ \\dot{x} ( t ) $的泛函。\n",
    "\n",
    "费曼观察到自由粒子的经典作用（V=0），\n",
    "$$\n",
    "S[b, a]=\\frac{m}{2}(\\dot{x})^2\\left(t_b-t_a\\right)=\\frac{m}{2} \\frac{\\left(x_b-x_a\\right)^2}{t_b-t_a},\n",
    "$$\n",
    "与自由粒子传播子(15.40)相关\n",
    "$$\n",
    "G(b, a) = \\sqrt{\\frac{m}{2 \\pi i\\left(t_b-t_a\\right)}} e^{i S[b, a] / \\hbar} .\n",
    "$$\n",
    "这就是量子力学和哈密顿原理之间备受追捧的联系。费曼随后提出了一种量子力学的重新表述，通过将$G(b, a)$表示为连接$a$和$b$的所有路径的加权和，将统计方面纳入其中，\n",
    "$$\n",
    "G(b, a)=\\sum_{\\text {路径 }} e^{i S[b, a] / \\hbar} \\quad \\text { （路径积分）。 } \\tag{45}\n",
    "$$\n",
    "这里，经典作用量S沿着不同的路径进行计算，作用量的指数在路径上进行求和。和（45）被称为路径积分，因为它在作用量$S[b, a]$上进行求和，每个作用量都是一个积分（在计算机上，积分和求和都是一样的）。经典力学和量子力学之间的本质联系是认识到在单位为$\\hbar \\simeq 10^{-34} \\mathrm{Js}$时，作用量是一个非常大的数字，$S / \\hbar \\geq 10^{20}$,因此，即使所有路径都进入和（45）中，主要贡献来自与经典轨迹$\\bar{x}$相邻的路径。事实上，由于S是经典轨迹的极值，它在路径变化的一阶是常数，因此附近的路径具有平滑且相对缓慢变化的相位。相比之下，远离经典轨迹的路径由快速振荡的$\\exp (i S / \\hbar)$加权，当包含许多路径时，它们倾向于相互抵消。在经典极限中，$\\hbar \\rightarrow 0$，只有单个经典轨迹起作用，并且（45）成为哈密顿最小作用量原理！\n",
    "\n",
    "束缚态波函数（理论）\n",
    "\n",
    "虽然你可能认为你已经看到了足够多的格林函数的表达式，但我们还需要另一个表达式来进行计算。让我们假设哈密顿算子 $\\tilde{H}$ 有一系列本征函数，\n",
    "$$\n",
    "H \\psi_n=E_n \\psi_n,\n",
    "$$\n",
    "每个解都标记为$n$。由于$H$是厄米特矩阵，这些解构成了一个完备的正交基，我们可以将一般解基于它们进行展开：\n",
    "$$\n",
    "\\psi(x, t)=\\sum_{n=0}^{\\infty} c_n e^{-i E_n t} \\psi_n(x), \\quad c_n=\\int_{-\\infty}^{+\\infty} d x \\psi_n^*(x) \\psi(x, t=0),\n",
    "$$\n",
    "其中，展开系数 $c_n$ 的值由 $\\psi_n$ 的正交性得出。如果我们将这个 $c_n$ 重新代入上面波函数展开式，我们得到恒等式\n",
    "$$\n",
    "\\psi(x, t)=\\int_{-\\infty}^{+\\infty} d x_0 \\sum_n \\psi_n^*\\left(x_0\\right) \\psi_n(x) e^{-i E_n t} \\psi\\left(x_0, t=0\\right) .\n",
    "$$\n",
    "与（39）比较，得出$G$的本征函数展开式：\n",
    "$$\n",
    "G\\left(x, t ; x_0, t_0=0\\right)=\\sum_n \\psi_n^*\\left(x_0\\right) \\psi_n(x) e^{-i E_n t} .\n",
    "$$\n",
    "我们将这和束缚态波函数联系起来（回想一下，我们的问题是计算它），通过（1）要求所有路径在空间位置$x_0=x$开始和结束，（2）取$t_0=0$，以及（3）对上式进行解析延拓到负虚时间（解析函数允许）：\n",
    "$$\n",
    "\\begin{aligned}G(x,-i \\tau ; x, 0) & =\\sum_n\\left|\\psi_n(x)\\right|^2 e^{-E_n \\tau}=\\left|\\psi_0\\right|^2 e^{-E_0 \\tau}+\\left|\\psi_1\\right|^2 e^{-E_1 \\tau}+\\cdots \\\\& \\Rightarrow\\left|\\psi_0(x)\\right|^2=\\lim _{\\tau \\rightarrow \\infty} e^{E_0 \\tau} G(x,-i \\tau ; x, 0) .\\end{aligned}\n",
    "$$\n",
    "这里的极限对应于长虚时间 $\\tau$, 在此之后，具有较高能量的部分 $\\psi$ 衰变得更快，只剩下基态 $\\psi_0$.\n",
    "\n",
    "上式方程直接根据传播子$G$给出了基态波函数的闭合解。虽然我们很快就会描述如何计算这个方程，但现在看看图\n",
    "![sec3-15.8.png](attachment:sec3-15.8.png)(图八)\n",
    "显示了一些计算结果。虽然我们从一个概率分布开始，该概率分布的峰值接近阱边缘的经典转折点，但在大量迭代后，我们最终得到一个类似于预期高斯的分布。在右边，我们看到一个通过关于经典轨迹$x(t)=A \\sin \\left(\\omega_0 t+\\phi\\right) $的统计变化生成的轨迹。\n",
    "\n",
    "\n",
    "15.8.2 格点路径积分（算法）\n",
    "\n",
    "\n",
    "因为在计算路径积分时，时间和空间是统一的，我们在时空内建立了一个离散点阵，并将粒子的轨迹可视化为一连串连接一个时间到下一个时间的直线\n",
    "![sec3-15.9.png](attachment:sec3-15.9.png)(图九)\n",
    "我们将点A和B之间的时间划分为大小为$\\epsilon$的N个等距步长，并用索引j标记它们：\n",
    "$$\n",
    "ε定义为 \\frac{t_b-t_a}{N} \\Rightarrow t_j=t_a+j \\varepsilon \\quad (j=0, N) .\n",
    "$$\n",
    "虽然使用轨迹在时间 $t_j$ 处的实际位置 $x\\left(t_j\\right)$ 来确定 $x_j$ 更为精确，但在实践中，我们均匀地对空间进行离散化，并将链接终止于最近的规则点。一旦我们有了格子，就很容易在链接上计算导数或积分：\n",
    "$$\n",
    "\\begin{aligned}\\frac{dx_j}{dt} & \\simeq \\frac{x_j-x_{j-1}}{t_j-t_{j-1}}=\\frac{x_j-x_{j-1}}{\\varepsilon} \\\\S_j & \\simeq L_j \\Delta t \\simeq \\frac{1}{2} m \\frac{\\left(x_j-x_{j-1}\\right)^2}{\\varepsilon}-V\\left(x_j\\right) \\varepsilon,\\end{aligned}\n",
    "$$\n",
    "其中我们假设拉格朗日函数在每条链路上都是恒定的。\n",
    "\n",
    "\n",
    "格点路径积分基于传播子的组合定理：\n",
    "$$\n",
    "G(b, a)=\\int d x_j G\\left(x_b, t_b ; x_j, t_j\\right) G\\left(x_j, t_j ; x_a, t_a\\right) \\quad\\left(t_a<t_j, t_j<t_b\\right) . \\tag{53}\n",
    "$$\n",
    "对于自由粒子，这产生\n",
    "$$\n",
    "\\begin{aligned}G(b, a) & =\\sqrt{\\frac{m}{2 \\pi i\\left(t_b-t_j\\right)}} \\sqrt{\\frac{m}{2 \\pi i\\left(t_j-t_a\\right)}} \\int d x_j e^{i(S[b, j]+S[j, a])} \\\\& =\\sqrt{\\frac{m}{2 \\pi i\\left(t_b-t_a\\right)}} \\int d x_j e^{i S[b, a]}\\end{aligned}\n",
    "$$\n",
    "其中我们将作用量进行了相加，因为线积分组合为 $S[b, j]+S[j, a]=S[b, a]$.对于图中的 $N$-链接路径，方程 (53) 变为\n",
    "$$\n",
    "G(b, a)=\\int d x_1 \\cdots d x_{N-1} e^{i S[b, a]}, S[b, a]=\\sum_{j=1}^N S_j,\n",
    "$$\n",
    "其中$S_j$是链接$j$的作用量。此时，图中所示的单路径积分已经变成一个$N$-项求和，随着时间步长$\\epsilon$接近零，它变成一个无穷求和。\n",
    "\n",
    "总而言之，费曼路径积分假设（45）意味着我们求和所有连接$A$和$B$的路径以获得格林函数$G(b, a)$.这意味着我们不仅要对一条路径中的链路求和，而且要对所有不同的路径求和，以产生哈密顿原理所要求的路径变化。该求和是受约束的，路径必须经过$A$和$B$，并且不能折返（因果关系要求粒子在时间上只能向前移动）。这就是路径积分的本质。因为我们在函数上以及沿着路径求和，该技术也被称为泛函积分。\n",
    "\n",
    "传播子 (45) 是连接 $A$ 和 $B$ 的所有路径之和，每条路径的权重是沿该路径作用的指数，明确地说：\n",
    "$$\n",
    "\\begin{aligned}G\\left(x, t ; x_0, t_0\\right) & =\\sum d x_1 d x_2 \\cdots d x_{N-1} e^{i S\\left[x, x_0\\right]}, \\\\S\\left[x, x_0\\right] & =\\sum_{j=1}^{N-1} S\\left[x_{j+1}, x_j\\right] \\simeq \\sum_{j=1}^{N-1} L\\left(x_j, \\dot{x}_j\\right) \\varepsilon,\\end{aligned}\n",
    "$$\n",
    "其中$L\\left(x_j, \\dot{x}_j\\right)$是时间$t=j\\epsilon$时链j上拉格朗日量的平均值。通过假设势V(x)与速度无关并且不依赖于其他x值（局部势），计算变得更加简单。接下来我们观察到，在基态波函数的表达式中，G以负虚时间来计算。因此，我们用$t=-i\\tau$来计算拉格朗日量：\n",
    "$$\n",
    "\\begin{gathered}\n",
    "L(x, \\dot{x})=T-V(x)=+\\frac{1}{2} m\\left(\\frac{d x}{d t}\\right)^2-V(x), \\\\\n",
    "\\Rightarrow L\\left(x, \\frac{d x}{-i d \\tau}\\right)=-\\frac{1}{2} m\\left(\\frac{d x}{d \\tau}\\right)^2-V(x) .\n",
    "\\end{gathered}\n",
    "$$\n",
    "我们看到，$L$中动能符号的逆转意味着$L$现在等于在实数正时间$t=\\tau$处求得的哈密顿量的负值：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "H(x, \\frac{d x}{d \\tau}) & =\\frac{1}{2} m\\left(\\frac{d x}{d \\tau}\\right)^2+V(x)=E, \\\\\n",
    "\\Rightarrow \\quad L\\left(x, \\frac{d x}{-i d \\tau}\\right) & =-H\\left(x, \\frac{d x}{d \\tau}\\right) .\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "这样，我们将$L$的$t$-路径积分重写为$H$的$\\tau$-路径积分，从而将作用量和格林函数表示为哈密顿量：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "S[j+1, j] & =\\int_{t_j}^{t_{j+1}} L(x, t) d t=-i \\int_{\\tau_j}^{\\tau_{j+1}} H(x, \\tau) d \\tau, \\\\\n",
    "\\Rightarrow \\quad G\\left(x,-i \\tau ; x_0, 0\\right) & =\\int d x_1 \\ldots d x_{N-1} e^{-\\int_0^\\tau H\\left(\\tau^{\\prime}\\right) d \\tau^{\\prime}},\n",
    "\\end{aligned}\n",
    "$$\n",
    "其中$H$在整个轨迹上进行线积分。接下来，我们将路径积分表示为每个链上粒子的平均能量$E_j=T_j+V_j$,然后对链求和，得到总能量：\n",
    "$$\n",
    "\\begin{gathered}\n",
    "\\int H(\\tau) d \\tau \\simeq \\sum_j \\varepsilon E_j=\\varepsilon \\mathcal{E}\\left(\\left\\{x_j\\right\\}\\right), \\\\\n",
    "\\mathcal{E}\\left(\\left\\{x_j\\right\\}\\right) \\stackrel{\\text { def }}{=} \\sum_{j=1}^N\\left[\\frac{m}{2}\\left(\\frac{x_j-x_{j-1}}{\\varepsilon}\\right)^2+V\\left(\\frac{x_j+x_{j-1}}{2}\\right)\\right] .\n",
    "\\end{gathered}\n",
    "$$\n",
    "\n",
    "在上式中，我们将每条路径链接近似为一条直线，使用欧拉微分规则来获得速度，并在每条链接的中点处计算势能。现在我们将这个 $G$ 代入我们的基态波函数解，其中空间中的初始点和终点是相同的：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim _{\\tau \\rightarrow \\infty} \\frac{G\\left(x,-i \\tau, x_0=x, 0\\right)}{\\int d x G\\left(x,-i \\tau, x_0=x, 0\\right)} & =\\frac{\\int d x_1 \\cdots d x_{N-1} \\exp \\left[-\\int_0^\\tau H d \\tau^{\\prime}\\right]}{\\int d x d x_1 \\cdots d x_{N-1} \\exp \\left[-\\int_0^\\tau H d \\tau^{\\prime}\\right]} \\\\\n",
    "\\Rightarrow\\left|\\psi_0(x)\\right|^2 & =\\frac{1}{Z} \\lim _{\\tau \\rightarrow \\infty} \\int d x_1 \\cdots d x_{N-1} e^{-\\varepsilon \\mathcal{E}} \\\\Z & =\\lim _{\\tau \\rightarrow \\infty} \\int d x d x_1 \\cdots d x_{N-1} e^{-\\varepsilon \\mathcal{E}}\n",
    "\\end{aligned}\n",
    "$$\n",
    "这些表达式与热力学的相似性，即使有配分函数Z，也不是偶然的；通过使量子力学的时间参数为虚数，我们将含时薛定谔方程转化为热扩散方程：\n",
    "$$\n",
    "i \\frac{\\partial \\psi}{\\partial(-i \\tau)}=\\frac{-\\nabla^2}{2 m} \\psi \\quad \\Rightarrow \\quad \\frac{\\partial \\psi}{\\partial \\tau}=\\frac{\\nabla^2}{2 m} \\psi .\n",
    "$$\n",
    "因此，格林函数中的路径求和，每条路径都按通常与热力学相关的玻尔兹曼因子$\\mathcal{P}=e^{-\\varepsilon \\mathcal{E}}$加权，这并不奇怪。我们通过将温度与时间步长倒数联系起来，使这种联系更加完整：\n",
    "$$\n",
    "\\mathcal{P}=e^{-\\varepsilon \\mathcal{E}}=e^{-\\mathcal{E} / k_B T} \\Rightarrow k_B T=\\frac{1}{\\varepsilon} \\equiv \\frac{\\hbar}{\\varepsilon} .\n",
    "$$\n",
    "因此，使时间连续的$\\epsilon \\to 0$极限是“高温”极限。投影基态波函数所需的$\\tau \\to \\infty$极限意味着，我们必须对虚时间较长的路径进行积分，即与典型时间相比，虚时间较长。正如我们在第一单元中对伊辛模型的模拟要求我们在系统达到平衡时等待很长时间，因此，目前的模拟要求我们等待很长时间，以便除了基态波函数之外的所有波函数都衰减。这就是我们找到基态波函数的解决方案。\n",
    "\n",
    "总之，我们已经将格林函数表示为路径积分，该路径积分需要沿着路径对哈密顿量进行积分，并对所有路径求和。我们将该路径积分作为时空格子中所有轨迹的总和进行计算。每个试验路径都以基于其作用量的概率出现，我们使用Metropolis算法将统计波动包含在链接中，就像它们处于热平衡状态一样。这类似于我们在第一单元中与伊辛模型的工作，但现在，我们不是根据能量的变化来拒绝或接受自旋翻转，而是根据能量的变化来拒绝或接受链接的变化。我们让算法运行的迭代次数越多，推导出的波函数就越需要平衡到基态。\n",
    "\n",
    "一般来说，蒙特卡洛格林函数技术最适合从正确答案的良好猜测开始，并让算法计算猜测的变化。对于目前的问题，这意味着如果我们从时空中的经典轨迹附近的路径开始，算法可能会很好地模拟经典轨迹周围的量子波动。然而，它似乎并不擅长从时空中的任意位置找到经典轨迹。我们怀疑后者是由于$\\delta S / \\hbar$太大，使得加权因子$exp（\\delta S / \\hbar）$剧烈波动（基本上平均为零），从而失去了其灵敏度。\n",
    "\n",
    "节约时间的小技巧\n",
    "\n",
    "当我们计算公式时，我们选择一个值 $x$，并在所有空间和时间上进行昂贵的线积分计算，以获得一个 $x$ 处的 $\\left|\\psi_0(x)\\right|^2$.为了获得另一个 $x$ 处的波函数，必须从头开始重复整个模拟。与其一次又一次地经历所有这些麻烦，我们将会一次性计算波函数的整个 $x$ 依赖性。诀窍是在概率积分中插入一个$\\delta$函数，从而将初始位置固定为 $x_0$, 然后对所有 $x_0$ 值进行积分：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "|\\psi_0(x)|^2 & =\\int d x_1 \\cdots d x_N e^{-\\varepsilon \\mathcal{E}\\left(x, x_1, \\ldots\\right)} \\\\\n",
    "& =\\int d x_0 \\cdots d x_N \\delta\\left(x-x_0\\right) e^{-\\varepsilon \\mathcal{E}\\left(x, x_1, \\ldots\\right)} .\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "该方程将波函数表示为所有路径上$\\delta$函数的平均值，这一过程可能完全不适合数值计算，因为在有限字长的计算机上表示奇异函数存在巨大的误差。然而，当我们模拟所有路径的总和时，总会有一些$x$值使得积分不为零，我们只需要累积各种（离散）$x$值的解，以确定所有$x$的$\\left|\\psi_0(x)\\right|^2$。\n",
    "\n",
    "为了理解这在实践中是如何工作的，考虑图中的路径$A B$，我们刚刚计算了它的总能量。我们通过让链上的一个点跳到点$C$（它改变了两个链接）来形成一条新路径。如果我们复制部分$A C$并将其用作扩展$A D$来形成顶部路径，我们看到路径$C B D$具有与路径$A C B$相同的总能量（作用），并且可以用于确定$\\left|\\psi\\left(x_j^{\\prime}\\right)\\right|^2$。在这种情况下，一旦系统达到平衡，我们通过将链接翻转为新值并计算新作用来确定新位置$x_j^{\\prime}$处的波函数的新值。某些$x_j$被接受越频繁，该点处的波函数就越大。\n",
    "\n",
    "格点上的实现\n",
    "\n",
    "\n",
    "编写程序，通过找到被积函数 $\\delta\\left(x_0-x\\right)$ 的平均值，并按照加权函数 $\\exp \\left[-\\varepsilon \\mathcal{E}\\left(x_0, x_1, \\ldots, x_N\\right)\\right]$ 分布路径，来计算积分 (45)。物理通过计算总能量 $\\mathcal{E}\\left(x_0, x_1, \\ldots, x_N\\right)$来进入.我们计算谐振子势的作用量积分\n",
    "$$\n",
    "V(x)= \\frac{1}{2} x^2\n",
    "$$\n",
    "对于质量为m=1的粒子。使用一组方便的自然单位，我们以1为长度单位，以1为时间单位。相应地，振荡器的周期为$T=2\\pi$。在图8这个计算中，我们从一个接近经典轨迹的初始路径开始，然后检查了关于这条路径的100万次变化。所有路径都被限制在x=1处开始和结束（结果略小于经典振荡的最大振幅）。\n",
    "\n",
    "当时间差$t_b-t_a$等于一个短时间，如$2 T$，系统还没有足够的时间达到基态平衡，波函数看起来像一个激发态的概率分布（几乎经典，粒子在速度为零的转折点附近的概率最高）。然而，当$t_b-t_a$等于更长的时间$20 T$时，系统有足够的时间衰减到基态，波函数看起来像预期的高斯分布。在任何一种情况下（图8右），轨迹通过时空涨落在经典轨迹附近波动。这种波动是Metropolis算法在搜索过程中偶尔上坡的结果；如果你修改程序，使搜索只下坡，时空轨迹将是一个非常平滑的三角函数（经典轨迹），但波函数作为经典轨迹波动的测量，将会消失！计算的明确步骤是\n",
    "1. 构建一个长度为 $\\varepsilon$ 的 $N$ 个时间步的网格（图9）。从 $t=0$ 开始，延伸到时间 $\\tau=N \\varepsilon$ [这意味着 $N$ 个时间间隔和时间上的 $(N+1)$ 个格点]。请注意，时间总是沿着路径单调递增。\n",
    "2. 构建一个由$M$个空间点组成的网格，这些空间点以大小为$\\delta$的步长分隔。使用比所使用的特征大小或范围大几倍的$x$值范围，并以$M \\simeq N$开始。\n",
    "3. 在计算波函数时，任何落在格点之间的 $x$ 或 $t$ 值都应该被分配给最近的格点。\n",
    "4. 将位置 $x_j$ 与每个时间 $\\tau_j$ 相关联，并满足初始和最终位置始终保持不变的边界条件，即 $x_N=x_0=x$。\n",
    "5. 选择一条直线连接对应经典轨迹的格点的路径。观察路径的连线的$x$值可能增加、减少或保持不变（与时间相反，时间总是增加）。\n",
    "6. 通过将路径上每个链接的动能和势能加起来，计算能量$\\mathcal{E}$，起点为$j=0$：\n",
    "$$\n",
    "\\mathcal{E}\\left(x_0, x_1, \\ldots, x_N\\right) \\simeq \\sum_{j=1}^N\\left[\\frac{m}{2}\\left(\\frac{x_j-x_{j-1}}{\\varepsilon}\\right)^2+V\\left(\\frac{x_j+x_{j-1}}{2}\\right)\\right] .\n",
    "$$\n",
    "7. 开始一系列重复步骤，其中与时间 $t_j$ 相关的随机位置 $x_j$ 变为位置 $x_j^{\\prime}$（图9 中的点 $C$）。这会改变路径中的两个链接。\n",
    "8. 对于改变的坐标，使用Metropolis算法用Boltzmann因子对变化进行加权。\n",
    "9. 对于每个格点，建立一个累加和，表示该点处波函数平方的值。\n",
    "10.  在每次单链变化（或决定不变化）后，将新的 $x$ 值的运行总和增加 1。在足够长的运行时间后，总和除以步骤数即为每个格点 $x_j$ 上 $\\left|\\psi\\left(x_j\\right)\\right|^2$ 的模拟值。\n",
    "11.  从不同的种子开始，重复整个链变化模拟。多次中等长度运行的平均波函数优于一次非常长的运行。"
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